
Class L_ A -545 

Book —^-"iiiS 

GoK'right'N" - 



COPyRIGIlT DEPOSU^ 



/ 
ELEMENTS 



OP 



SURVEYING- 



AND 



LEVELING. 



By CHARLES DAVIES, 'lL.D., 

AUTHOK OF A FULL COURSE OF MATHEMATICS. 



REVISED BY 

J. HOWARD VAN AMRINGE, A.M., Ph.D, 

PROFESSOR OP MATHEMATICS IN COLUMBIA COLLEGE. 



NEW YORK ■:• CINCINNATI •:. CHICAGO 

AMERICAN BOOK COMPANY 






a^^oi>__ 

DAVIESS MATHEMATICAL SERIES 

For Elementary Schools 

Davies's Primary Arithmetic 
Davies's Intellectual Arithmetic 
Davies's First Book in Arithmetic 
Davies's Standard Arithmetic 
Davies's Practical Arithmetic 
Davies's Complete Arithmetic 

For Secondary Schools 

Daviess University Arithmetic 
Davies's New Elementary Algebra 
Davies's Bourdon's Algebra 
Davies's Elementary Geometry and 

Tri2:onometry 
Davies's^ Legendre's Geometry and 

Trigonometry 

For Co.lleges and Advanced Students 
Davies's University Algebra 
Davies's Analvtical Geometry 
Davies's Analytical Geometry and 

Calculus 
Davies's Descriptive Geometry 
Davies's Elements of Surveying 












JUL -5 1898 J 



Twoirr 



IVED. 



^^ C^ ^^ ^ ^^,^4 



PREFACE. 



DAVIES' Elements of Surveying, first published in 1830, 
was designed as a text-book for the pupils of the U. S. 
Military Academy, and in its preparation little regard was had to 
the supposed wants of other institutions. 

The work was, however, received by the public with more 
favor than was anticipated, and soon became a leading text-book 
in Colleges, Academies, and the higher grade of Schools. For 
the purpose of adapting it more fully to the requirements of 
these institutions, the autlior made many changes in successive 
editions, and gave it his final revision in 1870. 

In the present edition, while the admirable features which 
have hitherto commended the work so highly to institutions of 
learning and to practical surveyors have been retained, some of the 
topics have been abridged in treatment and some enlarged, others 
have been added, and the whole has been arranged in the order of 
progressive development. 

It has been the intention to begin with the very elements of 
the subject and to combine those elements in the simplest man- 
ner, so as to render the higher branches of Plane Surveying com- 
paratively easy. The necessary principles of logarithms and 
plane trigonometry are given, and their mode of application 
shown. All the instruments needed for plotting have been 
carefully described ; and the uses of those required for the 
measurement of angles are fully explained. 

In the section on Magnetic Declination or Variation of the 
Needle, papers of the U. S. Coast and Geodetic Survey have been 
largely used. From them have been taken : — tables of annual 
changes in declination, and for computing the declination at any 
epoch, at various places in the United States, which will be found 



IV PREFACE. 

of especial value in re-running lines of old surreys ; also, new 
tables of the times and azimuths of Polaris when at elonsration, 
for use in determining the true meridian with compass or tran- 
sit, which with the rules given for interpolation are more accurate 
than any similar tables previously published. 

A full account is given of the system adopted in the survey of 
the public lands; and although the method is simple, it has, 
nevertheless, been productive of great results, by defining, with 
mathematical precision, the boundaries of lands in the new States, 
and thus settling their titles on an indisputable basis. In this 
connection official instructions and diagrams issued by the U. S. 
General Land Ofi&ce have been used, and, as the principal lines of a 
government survey must be run with reference to the true meridian, 
the Solar Compass and solar attachment to transit are described. 

A change made in the present edition, which must prove par- 
ticularly acceptable, is the transformation of the article on Mining 
Surveying into a complete treatise, in which the location of claims 
on the surface, the latest and best methods of underground travers- 
ing, etc., the calculation of ore-reserves, and all that pertains to the 
work of the Mining Surveyor, are fully explained, and illustrated 
by practical examples. This improvement is due, substantially, to 
John G. Murphy, Esq., E.M., at one time Territorial Geologist of 
Wyoming Territory, an expert Mining Engineer of large and 
varied practice. 

In addition to acknowledgments elsewhere made, the under- 
signed is indebted to Professor Eodney G. Kimball, of the Brook- 
lyn Polytechnic Institute, for valuable suggestions and labor, and 
to the Messrs. W. and L. E. Gurley, of Troy, N. Y., for their 
courtesy in furnishing him, for use, advance sheets of the last 
edition of their Instrument Manual and many cuts of their 
surveying instruments. 

J. H. VAN" AMRLNTGE, 

Editor of Davies' Cotjbsb op Mathekatics. 
CoLTjMBiA College, N. T., ) 
SepteTnber, 1883. S 



CONTENTS. 



BOOK I. 

INTRODUCTORY PRINCIPLES AND DEFINITIONS. 

SECTION PAGE 

I. — Logarithms 9-16 

II.— Plane TRiGONOMETrvY 17-32 

III. — Instruments for Plotting 33-39 

IV. — General Definitions 40-41 



BOOK II. 

CHAIN SURVEYING. 

I. — ^Measurement of Distances 43-58 

Necessary instruments 43 

To measure a horizontal line 44 

Applications 48 

Standard of measure 57 

II. — Area or Contents of Ground 58-70 

BOOK III. 

COMPASS SURVEYING. 

I. — Definitions 71-73 

II. — Surveyor's Compass 74-78 

III. — Work on the Field 79-90 

Field notes 79 

Necessary measurements on the field 80 

Errors of compass 80 

Measurements by off-set courses 82 

General example 84 

To correct local attraction 89 



VI CONTENTS. 

SECTION PAGE 

IV. — Area or Contents of Ground 90-131 

Traverse table and its uses 90 

Balancing tlie work 94 

Double meridian distances 97 

Area 99 

Plotting - 102 

Examples 104 

To supply omissions \n field notes Ill 

Anotlier mode of finaing areas 120 

V. — Magnetic Declination or Variation of the Needle. . .131-152 

Definitions 131 

Daily variatior (with table) 132 

Secular variation (with tables) 134 

Method of finding declination (with tables) 140 

To find true meridian with compass 145 

Vernier compass and its use 150 

Form of survey bill 152 

BOOK IV. 

TRANSIT SUEVEYING. 

I. — Surveyor's Transit 153-165 

II. — Measurement of Angles 166-183 

Horizontal and vertical angles 166, 167 

Azimuths and bearings 168, 169 

To find true meridian with transit 172 

Applications to heights and distances 174 

III.— Ranging out Lines, Etc 184-189 

To range out a line 184 

Measurement of distances by transit 186 

To survey a road, boundary of estate, etc 189 

To survey streets of a town or city 189 

IV. — Farm Surveying by Transit 190-195 

BOOK V. 

LAYING OUT AND DIVIDING LAND. 

I.— Of Dividing Land 196-203 

II. — Public Lands of the United States .... .203-211 



CONTENTS. vii 

BOOK VI. 

TRIGONOMETRICAL SURVEYING. 

SECTION PAGE 

I.— Making the Survey 212-283 

Definitions and general remarks 212 

Base line 215 

Signals 217 

Heliotrope 218 

Theodolite 219 

Measurement of angles, and notes 221 

Reduction to the centre 226 

Survey of a harbor 228 

II. — Filling up the Survey 233-240 

By the compass 233 

By the plane table 234 

III. — Plotting the Triangulation 240-245 

BOOK VII. 

LEVELING. 

I. — Definitions and Principles 246-248 

II. — Instruments 249-257 

Y Level 249 

Leveling rods 255 

Tests of adjustment 258 

III. — Leveling in the Field 258-262 

IV.— Section Leveling 263-276 

Definitions and principles 263 

Drawing the profile ... 270 

Establishment of the grade 270 

Examples 276 

V. — Cross-Section Leveling 277-288 

Slopes 277 

Setting slope stakes 278 

VI. — Computation of Earthwork 288-296 

BOOK VIII. 

TOPOGRAPHICAL SURVEYING. 

Definition and principles 297 

Examples and plotting 298 



Vlll COK'TEiq'TS. 

8KCTI0N P^ej. 

Shading and delineation 310 

Topographical signs, U. S. Coast Survey 313 

BOOK IX. 

RAILWAY CURVES. 

Definitions and principles 317 

Location of curves 320 

Laying oflf ordinates 327 

Reversed and compound curves 329 

BOOK X. 

MININ"G SURVEYING. 

I. — Definitions and General Principles 338-340 

IL — Method of Locating Claims 340-345 

III. — Underground Traversing, Etc 346-361 

To make the traverse 346 

To reduce the traverse 350 

To plot traverse on the surface 353 

To plot traverse on paper 360 

IV. — Practical Applications 361-374 

Problems 361 

Example of a developed mine 366 

Calculation of ore-reserves 370 

Deposit mine with mill connections 372 

APPENDICES. 

A. — Solar Compass and Solar Attachment to Transit 1-15 

B.— Sextant 16-20 

C. — Instructions to U. S. Mineral Surveyors 21-29 

TABLES. 

Logarithms of Numbers 1-16 

Logarithmic Sines and Tangents 17-62 

Natural Sines 63-71 

Traverse Table 72-161 



BOOK I. 

INTRODUCTORY PRINCIPLES AND DEFINITIONS. 



SECTION I. 

LOGARITHMS. 

1. The logarithm of a given number is the exponent of the 
power to which it is necessary to raise a fixed number to produce 
the given number. 

The fixed mimher is called the base of the system. In the 
common system, to which alone reference is made in this section, 
the base is 10. Every number is, therefore, regarded as some 
power of 10, and the exponent of that power is the logarithm of 
the number. 

2. If a number is an exact power of 10, its logarithm is a 
whole number. If a number is not an exact power of 10, its 
logarithm is composed of two parts, a whole number called the 
Characteristic, and a decimal part called the Mantissa. 
Thus, 225 being greater than 10^ and less than 10^, its logarithm 
is found to be 2.352183, of which 2 is the characteristic and 
.352183 is the mantissa. 

3. In a table of logarithms, the mantissas only are necessarily 
given. The characteristic of the logarithm of a number is deter- 
mined by one of the two following rules : 

Rule I. — The characteristic of the logarithm of any 
whole number is positive, and numerically 1 less than the 
number of places of figures in the given number » 



iO ELEMENTS OF SURVEYING. [BOOK I. 

Rule II. — The characteristic of the logarithm of a 
decimal fraction is negative, and nuinerically 1 greater 
than the number of O's that immediately follow the 
decimal point. 

Note 1. —In the logarithm of a decimal fraction, the charac- 
teristic alone is negative, the mantissa being always positive. 
This fact is indicated by writing the negative sign over the 
characteristic: thus, 2. 371465, is equivalent to — 2 -f .371465. 

l^OTE 2. — It is to be observed, that the characteristic of the 
logarithm of a mixed number is the same as that of its entire 
part. Thus, the characteristic of the logarithm of 725.4275 is 
the same as the characteristic of the logarithm of 725. 

4. A Table of Logarithms is a table by means of which 
may be found the logarithm corresponding to any number, or the 
number corresponding to any logarithm. 

In the table appended, the mantissas alone are given; the 
characteristic may be found by one of the rules of Art. 3. 

The mantissa of the logarithm of any number is not changed 
by multiplying or dividing the number by any exact power of 10. 
Hence, in finding the mantissa of the logarithm of a number, the 
position of the decimal point may be changed at pleasure. Thus, 
the mantissa of the logarithm of 456357, is the same as that of 
the number 4563.57 ; and the mantissa of the logarithm of 75, is 
the same as that of 7500. 

6. To find the logarithm of a number between 1000 and 
10,000. — Find the characteristic by the first rule of Art. 3. 
To determine the mantissa, find in the column headed ^'N" the 
left-hand three figures of the given number ; then pass along the 
horizontal line in which these figures are found, to the column 
headed by the fourth figure of the given number, and take out 
the four figures found there ; pass back again to the column 



SEC. I.] LOGARITHMS. 11 

headed "0/' and there will be found in this column, either upon 
the horizontal line of the first three figures or a few hues above 
it, a number consisting of six figures, the left-hand two figures of 
which must be prefixed to the four already taken out. Thus, 

Log 8979 = 3.953228. 

If, however, any dots are found at the place of the four figures 
first taken out, or if in returning to the " " column any dots 
are passed, the two figures to be prefixed are the left-hand two of 
the six figures of the *' " column immediately below. Dots in 
the number taken out must be replaced by zeros. Thus, 

Log 3098 = 3.491081 
Log 2188 = 3.340047 

6. To find the logarithm of a number between 1 and 1000. — 

Find the characteristic by the first rule of Art. 3. To find the 
mantissa, fill out the given number to four places of figures (or 
conceive it to be so filled out) by annexing O's (see Art. 4), and 
find the mantissa corresponding to the resulting number, as in 
Art. 5. Thus, to find log. of 75 : characteristic is 1, by the rule ; 
the mantissa is the same as that corresponding to 7500, i. e., 
.875061; hence. 

Log 75 = 1.87506L 
In the same way, 

Log 2 = 0.301030. 

7. To find the logarithm of a number greater than 10,000. — 
Find the characteristic by the first rule of Art. 3. To find the 
mantissa : set aside all of the given number except the left-hand 
four figures, aud find the mantissa corresponding to these four, 
as in Art. 5 ; multiply the corresponding tabular difference, 
found in column ^^D," by the part of the number set aside, and 
discard as many of the right-hand figures of the product as there 
are figures in the multiplier, and add the result thus obtained to 



13 ELEMENTS OF SUKYETIi^G. [BOOK I. 

the mantissa already found. If the left-hand figure of those dis- 
carded is 5, or more, increase the number added by 1. 

Note. — It is to be observed that the tabular differ encey found 
in column ^^D," is millionths, and not a vhole number; and 
that, therefore, the result to be added *^to the mantissa already 
found " is millionths. 

Example. — To find the logarithm of 672887: the character- 
istic is 5 ; set aside 87, and the mantissa corresponding to 6728 is 
.827886 ; the corresponding tabular difference is 65, which mul- 
tiplied by 87, the part of the number set aside, gives 5655 ; as 
there are two figures in the multiplier, discard the right-hand 
two figures of this product, leaving 56 ; but as the left-hand 
figure of those discarded is 5, call the result 57 (which is 
millionths) ; adding this 57 to the mantissa already found, will 
give .827943 for the required mantissa ; hence, 

Log 672887 = 5.827943. 
In the same way. 

Log 3710053 = 6.569380. 

8. To find the logarithm of a decimal. — Find the character- 
istic by the second rule of Art. 3. To find the mantissa, drop 
the decimal point and consider the decimal a whole number. 
Find the mantissa of the logarithm of this number as in preced- 
ing articles, and it will be the mantissa required. Thus, 

Log .0327 = 2.514548 
Log .378024 = L 577520. 

IN'OTE. — To find the logarithm of a mixed number, find the 
characteristic by Note 2, Art. 3 ; then drop the decimal point 
and proceed as above. 

9. To find the number corresponding to a given logarithm. — 

The rule is the reverse of those just given. Look in the table for 
the mantissa of the given logarithm. K it cannot be found, take 



SEC. I.] LOGARITHMS. 13 

out the next less mantissa, and also the corresponding number, 
which set aside. Find the difference between the mantissa taken 
out and that of the given logarithm ; annex any number of O's, 
and divide this result by the corresponding number in the column 
"D." Annex the quotient to the number set aside, and then, if 
the characteristic is positive, point off, from the left hand, a num- 
ber of places of figures equal to the characteristic plus 1 ; the 
result will be the number required. 

If the characteristic is negative, prefix to the figures obtained 
a number of O's one less than the number of units in the nega- 
tive characteristic and to the whole prefix a decimal point ; the 
result, a pure decimal, will be the number required. 

Example. — Let it be required to find the number correspond- 
ing to the logarithm 5.233568. 

The next less mantissa in the table is 233504 ; the correspond- 
ing number is 1712, and the tabular difference is 253. 

OPEKATION. 

Given mantissa, 233568 

Next less mantissa, . . . . 233504 . . 1712 

253 ) 6400000 ( 25296 

.-. The required number is 171225.296. 

The number corresponding to the logarithm 2.233568 is 
.0171225+. 

10. Multiplication by Logarithms. — Rule. — Find the 
logarithms of the factors and take their sum; then find 
the number corresponding to the resulting logarithm, and 
it will he the product required. 

Example. — Find the continued product of 3.902, 5971.6, 
and .0314728. 



14 ELEMENTS OF SUEVErii^G. [BOOK I. 

OPERATION. 

log 3.902 = 0.591287 
log 5971.6 = 3.776091 
log .0311728 = 2.197936 

2.8653U .♦. 733.354, product 

Here, the 1 carried added to the 3 gives 4, which added to — 2 
gives 2 as the characteristic of the logarithm of the product. 

11. DiTision by Logarithms. — Rule. — Find the loga- 
rithms of til e dividend and divisor, and sitbtract the latter 
from the fomner; then find the number corresponding to 
the resulting logarithm, and it udll he the quotient re- 
quired, 

EXAMPLES. 

1. Divide 24163 by 4567. 



log 24163 

log 4567 



OPEEATIOX, 

. 4.383151 

. 3.659631 



0.723520 .-. 5.29078, quotient. 



2. Divide 0.7438 by 12.9476. 



log 0.7438. 
log 12.9476 . 



0PEBATI02f . 

. T. 871456 
. 1.112189 



2.759267 



0.057447, quotient 



Here, 1 taken from I, gives 2 for a result. The subtraction, 
as in this case, is always to be performed in the algebraic sense. 

The operation of division, particularly when combined with 
that of multiplication, can often be simplified by using the 
principle of the 



SEC. I.] LOGAKITHMS. 15 

12. Arithmetical Complement. — The aritlimetical comple- 
ment of a logarithm is the remainder obtained by subtracting it 
from 10. Thus, 8.130456 is the arithmetical complement of 
1.869544. The arithmetical complement is denoted by the 
symbol (a. c). 

The following is the rule for the use of the arithmetical com- 
plement in division by logarithms: 

Rule. — Find the logaritlnn of the dividend, and the 
arithmetical complement of the logarithm of the divisor, 
add them together, and diminish the sum by 10 ; the 
nuwyher corresponding to the resulting logarithjn will be 
the quotient required. 

Examples.— 1. Divide 37.149 by 5^3.76. 
log 37.149 . . . 1.569947 
(a. c.) log 523.76 . . . 7.2808 67 

2.85081 4 .-. 0.0709273, quotient 

The operation of subtracting 10 is performed mentally. 

2. Find x in the proportion, 

602.647 : 2.29863 w x : .037293. 
log 602.647 . . 2.780063 
(a. c.) log 2.29863 . . 9.638531 
log .037293 . . 2.57162 7 
log :c . . . . 0.99022 1 .-. a; = 9.7773 + . 

3. Divide the product of 3.58884 and 5672, by the product of 
89721 and 42.056. 

log 358884 . . . 5.554954 

log 5672 .... 3.753736 

(a. c.) log 89721 . . . 5.047106 

(a. c.) log 42.056 . . . 8.376182 

2. 731978 . • . 539.48, result. 
20 is here subtracted, as (a. c.) has been twice used. 



16 ELEMEI^TS OF SURVEYING. [BOOK I. 

Note. — If the logarithm, whose arithmetical complement is 
taken, exceeds 10, subtract it from 20, and reject 20 in the final 
operation. 

13. Raising to Powers by Logarithms.— Eule.—2^7i^ 
the logarithm of the number, and inultiply it by the 
exponent of the power ; then find the number correspond- 
ing to the resulting logarithm, and it will he the power 
required. 

Example. — Find the 5th power of 9. 

log 9 0.954248 

5 

4.771215 .-. 59049, power. 

14. Extracting Roots by Logarithms.— Rule.— i^iw,^ 
the logarithm of the number and divide it by the index 
of the root; then find the number corresponding to the 
resulting logarithm, and it will be the root required. 

Example. — Find the cube root of 4096. 

The logarithm of 4096 is 3.612360, and one-third of this is 
1.204120. The corresponding number is 16, which is the root 
sought. 

If the characteristic of the logarithm of the given number is 
negative and not exactly divisible by the index of the root, add 
to it such negative quantity as shall make it exactly divisible, and 
add also to the mantissa a numerically equal positive quantity. 

Thus, let the square root of .00863 be required. 

log .00863 = 3.936011 = 4 + 1.936011. 
4 + 1.936011 



log v. 00863 = "^ • = 2.968006. 

The number sought is therefore .09289 + . 



SEC. II.] 



PLANE TRIGONOMETRY. 



17 



SECTION II. 

PLAN E TRIGO NO M ETRY. 

15. Plane Trigonometry is that branch of Mathematics 
which treats of the solution of plane triangles. 

In every plane triangle there are six parts : three sides and 
three angles. When three of these parts are given, one being a 
side, the remaining parts may be found by computation. The 
operation of finding the unknown parts, is called the solution 
of the triangle. 

16. A plane angle is measured by the arc of a circle included 
between its sides, the centre of the circle being at the vertex, 
and its radius being equal to 1. 

Thus, if the vertex A be taken as a 
centre, and the radius AB hQ equal to 1, 
the intercepted arc BG will measure the 
angle A. 

Let ABCD represent a circle whose radius is equal to 1, 
and AC, BD, two diameters perpendicu- 
lar to each other. These diameters divide 
the circumference into four equal parts, 
called quadrants; and because each of 
the angles at the centre is a right angle, 
it follows that a right angle is measured 
by a quadrant. An acute a7igle is meas- 
ured by an arc less than a quadrant, 
and an obtuse angle, by an arc greater than a quadrant. 

17. In Geometry, the unit of angular measure is a right 
angle ; so in Trigonometry, the primary unit is a quadrant, 
which is the measure of a right angle. 




Fig. 1. 





18 ELEMEi^TTS OF SURVETIl^Q. [BOOK I. 

For convenience, the quadrant is divided into 90 equal parts, 
each of which is called a degree ; each degree into 60 equal parts, 
called minutes; and each minute into 60 equal parts, called 
seconds. Degrees, minutes, and seconds, are denoted by the 
symbols °, ', ". Thus, the expression 7° 22' 33", is read, 
7 degrees, 22 minutes, and 33 seconds. Fractional parts of a 
second are expressed decimally. 

18. The complement of an arc is the difference between that 
arc and 90°. The complement of an 
angle is the difference between that 
angle and a right angle. 

Thus, EB is the complement of 
AE, and FB is the complement of 
CF. In like manner, FOB is the 
complement of AGE, and FOB is 
the complement of COF. ^^^ 

In a right-angled triangle, the acute angles are complements 
of each other. 

19. The supplement of an arc is the difference between that 
arc and 180°. The su2)plement of an angle is the difference 
between that angle and two right angles. 

Thus, EC (Fig. 3) is the supplement of AE, and FG the 
supplement of AF. In like manner, EOC is the supplement 
of A OF, and FOC the supplement of A OF. 

In any plane triangle, either angle is the supplement of the 
sum of the other two. 

20. Instead of the arcs themselves, certain functions of the 
arcs, as explained below, are used. A function of a quantity is 
something which depends upon that quantity for its value. 

The following functions are the only ones needed for solving 
triangles ; 



SEC. II.] 



PLANE TRIGONOMETRY. 



19 




21. The sine of an arc is the distance of one extremity of 
the arc from the diameter through the other extremity. 

Thus, PM (Fig. 4) is the sine 

of AM, and P'M' is the sine of '^Z s t' 

AM'. 

22. The cosine of an arc is 
the sine of the complement of 
the arc, ^^ complement sine" be- 
ing contracted into cosine. 

Thus, NM (Fig. 4) is the co- 
sine of AM, and NM' is the 

cosine of AM'. These lines are respectiyely equal to OP 
and OP'. 

23. The tangent of an arc is the perpendicular to the radius 
at one extremity of the arc, limited by the prolongation of the 
diameter through the other extremity. 

Thus, AT (Fig. 4) is the tangent of the arc AM, and AT'" 
is the tangent of the arc AM'. 

24. The cotangent of an arc is the tangent of its complement, 
" complement tangent " being contracted into cotangent. 

Thus, BT' (Fig. 4) is the cotangent of the arc AM, and 
BT" is the cotangent of the arc AM'. 

The sine, cosine, tangent, and cotangent of an arc, a, are, 
for convenience, written sin a, cos a, tan a, and cot a. 

These functions of an arc may also be considered as func- 
tions of the angle which the arc measures. 

Thus, (Fig. 4) PM, NM, AT, and BT', are respectively the 
sine, cosine, tangent, and cotangent of the angle A OM, as well 
as of the arc AM. 

25. The sine of an arc is equal to the sine of its supplement ; 
and, in general, any function of an arc is equal to the corres- 



20 ELEMENTS OF SUKVEYING. [BOOK I. 

ponding function of its supplement. Thus, if A is any arc 
or angle, 

sin A = sin (180° — A) ; 

cos A = cos (180° — A) ; 

tan A = tan (180° — A) ; 

cot A = cot (180° — A). 

Note. — These relations exist between the numerical values 
of the functions ; the algebraic signs, which they have in the 
different quadrants, are not considered. 

26. A Natural Sine, Cosine, Tangent, or Cotangent, 
is the sine, cosine, tangent, or cotangent, of an arc whose 
radius is 1. 

A Table of Natural Sines is a table from which the 
natural sine, cosine, tangent, or cotangent of any arc may 
be found. 

The Table of Natural Sines, beginning at page 63 of the 
tables, gives the values of the sines and cosines only. If the 
tangent or cotangent of an arc. A, is desired, it may be found 
by the relation, 

sin ^ , . cos A 

tan A = J ; cot ^ = j* 

cos A sm A 

TABLE OF LOGARITHMIC SINES. 

27. A Logarithmic Sine, Cosine, Tangent, or Cotan- 
gent is the logarithm of the sine, cosine, tangent, or cotangent 
of an arc whose radius is 10,000,000,000. 

A Table of Logarithmic Sines and Tangents is a table 
giving the logarithm of the sine and cosine, tangent and cotan- 
gent of any arc or angle. 

The logarithm of the tabular radius is 10. 



SEC. II.] PLANE TRIGONOMETRY. 21 

Any logarithmic function of an arc or angle may be found by 
multiplying the corresponding natural function by 10,000,000,000, 
and then taking the logarithm of the result ; or more simply, by 
taking the logarithm of the corresponding natural function, and 
then adding 10 to the result. 

28. In the table, beginning at page 18 of the tables, the 
logarithmic functions are given for every minute from 0° to 90°. 
In addition, their rates of change for each second, are given in 
the column headed " D." 

For the sine and cosine, there are separate columns of dif- 
ferences, which are written to the right of the respective 
columns; but for the tangent and cotangent, there is but 
a single column of differences, which is written between 
them. 

The angle obtained by taking the degrees from the top of the 
page, and the minutes from any line on the left hand of tlie 
page, is the complement of that obtained by taking the degrees 
from the lottom of the page, and the minutes from the same line 
on the right hand of the page. But, by definition, the cosine and 
the cotangent of an arc are, respectively, the sine and the 
tangent of the complement of that arc (Arts. 22 and 24) ; 
hence, the columns designated sine and tang, at the top of the 
page, are designated cosine and cotang at the bottom. 

29. To find the logarithmic functions of an angle which is 
expressed in degrees and minutes. 

If the angle is less than 45°, look for the degrees at the top 
of the page, and the minutes in the left hand column; then 
follow the corresponding horizontal line to the column desig- 
nated at the top by sine, cosine, tang, or cotang, as the case 
may be ; the number there found is the logarithm required. 
Thus, 



22 ELEMENTS OF SURVETIXG. [BOOK I. 

log sin 19° 55' . . . 9.532312 
log tan 19° 55' . . . 9.559097 

If the angle is greater than 45°, look for the degrees at the 
bottom of the page, and for the minutes in the right hand 
cohimn ; then follow the corresponding horizontal line back- 
wards to the column designated at the hottom by sine, cosine, 
tang, or cotang, as the case may be ; the number there found is 
the logarithm required. Thus, 

log cos 52° 18' . . . 9.786416 
log tan 52° 18' . . . 10.111884 

30. To find the logarithmic functions of an angle which is 
expressed in degrees, minutes, and seconds. 

Find the logarithm corresponding to the degrees and minutes 
as before ; then multiply the corresponding number taken from 
the column headed '"D" (which is millionths), by the number of 
seconds, and add the product to the preceding result, for the 
sine or tangent, and subtract it therefrom for the cosine or 
cotangent. 

EX A M P L E S . 

1. Find the logarithmic sine of 40° 26' 26". 

OPERATION. 

log sin 40° 26' 9.811952 . 

Tabular difference 2.47 
No. of seconds . 28 

Product. . . . 69.16 to be added . ^ 

log sin 40° 26' 28" 9.812021 

The same rule is followed for the figures discarded (in this 
case 16), as in Art. 7. 



SEC. il] plake tkigonometry. 23 

2. Find the logarithmic cosine of 53° 40' 46". 

OPERATION. 

log cos 53° 40' 9.772675 

Tabular difference 2.86 

No. of seconds . 46 

Product . . . 131.56 to be subtracted 132 

log cos 53° 40' 46" 9.772543 

If the angle is greater than 90°, we find the required function 
of its supplement (Art. 25). 

3. Find the logarithmic tangent of 118° 18' 25". 

OPERATION. 

180° 
Given arc . ... . . 118° 18' 25" 

Supplement 61° 41' 35" 

log tan 61° 41' ' . . . . 10.268556 

Tabular difference 5.04 

No. of seconds . 35 

Product . . . 176.40 to be added . 176 

log tan 118° 18' 25" 10.268732 

31. To find the angle corresponding to any logarithmic 

function. 

This is done hy reversing the preceding rule : Look in the 
proper column of the table for the given logarithm ; if it is 
found there, the degrees are to be taken from the top or bottom, 
and the minutes from the left or right hand column, as the case 
may be. If the given logarithm is not found in the table, then 
find the next less logarithm, and take from the table the correc- 
ponding degrees and minutes, and set them aside. Subtract the 
logarithm found in the table, from the given logarithm, annex 



24 ELEMENTS OF SURVEYING. [BOOK i. 

two O's to the remainder, and divide this result by the corres- 
ponding tabular difference. The quotient will be seconds, which 
must be added to the degrees and minutes set aside, in thei^ase 
of a sine or tangent, and subtracted, in the case of a cosine or a 
cotangent. 

EXAMPLES. 

1. Find the angle corresponding to the logarithmic sine 
9.422248. 

OPEKATION. 

Given logarithm . . . 9.422248 
Next less in table . . . 9.421857 ... 15° 19' 
Tabular difference 7.68 ) 391.00 ( 51", to be added. 

Hence, the required arc is 15° 19' 51". 

2. Find the angle corresponding to the logarithmic cosine 
9.427485. 

OPERATION. 

Given logarithm . . . 9.427485 
Next less in table . . . 9.427354 . . . 74° 29' 
Tabular difference 7.58 ) 131.00 ( 17", to be subt. 

Hence, ihe required angle is 74° 28' 43". 

32. Theorem I. — The sides of a plane triangle are propor^ 
tional to the sines of their opposite angles. 

In the triangle ABC 
let the large letters A, B, 
(7, designate the angles, 
and the corresponding 
small letters, a, h, c, the ^" 

Fig. 5. 

sides opposite ; then, 

« : 5 : : sin ^ : sin ^ ; 

a '. c : : sin ^ : sin C ; 

b : c : : sin ^ : sin C. 




SEC. II. J 



PLANE TKIGOKOMETKY. 



25 



33. Theorem II. — In any plane triangle, the sum of the 
two sides containing any angle, is to their difference, as the 
tangent of half the sum of the two other angles is to the tangent 
of half their difference. 

Thus, in the triangle ABG (Fig. 5), 

a + h : a—h :: imn^ (A-\-B) : tan|-(^— ^); 
a-\-c : a—c :: tan ^ {A-{-C) : tan ^ (A — C); 
Ij^c : h—c :: tani(^+C) : tan|(^— C). 

By solving any one of the above proportions, the first three 
terms being known, the tangent of half the difference of the two 
unknown angles is obtained, and from this tangent the half 
difference itself is found. The greater of the two unknown 
angles is equal to half their sum added to half their difference ; 
the smaller is equal to half their sum diminished by half 
their difference. 



34. Theorem III. — In any plane triangle, if a line is 
drawn from the vertex of the vertical angle perpendicular to 
the base, dividing it into two segments : then, the sum of the two 
segments, or the whole hase, is to the sum 

of the two other sides, as the difference of 
those sides, to the difference of the segments. 

Thus, in the triangle ABG (Fig. 6), 
s-\-s' : b-i-c :: b—c: s—s', 

35. Theorem IV. — In any right-angled plane triangle, 
radius is to the tangent of either of the acute angles, as the side 
adjacent to the side opposite. 

Let CAB (Fig. 7) be the proposed triangle, and denote the 
radius by R : then will 




^6 ELEMENTS OF SURVETINa 

B* : tan C :: ACy or b : AB, or c ; 
Also, 

E* : tan B : : AB, or c : AC, or b, ^^ ^ 

36. Theorem Y. — /?i an?/ right-angled triangle, radius is 
to the cosine of either of the acute angles, as the hypothenuse to 
the side adjacent. 

In the triangle CAB (Fig. 7), 

i^* : cos C : : BC, ov a '. AC, ov b', 
Also, 

i2* : cos B : : BC, or « : ^^, or c. 

37. Solution of Triangles. — The relations between the 
sides and angles of plane triangles, stated in these five theorems, 
are sufficient to solve all the cases of Plane Trigonometry. Of 
the six parts, which make up a plane triangle, three must be 
given, and at least one of these must be a side, before the others 
can be determined. 

If the three angles only are given, it is plain that an indefinite 
number of similar triangles may be constructed, the angles of 
which may be respectively equal to the angles that are given 
and, therefore, the sides could not be determined. 

Assuming, with this restriction, any three parts of a triangle, 
one of the four following cases will always be presented: 

I. When two angles and a side are given. 
II. When two sides and an angle opposite one of them 
are given. 

III. When two sides and the included angle are given. 

IV. When the three sides are given. 



* If logaHthmic functions are used, R is equal to 10,000,000,000, and its logariihm is 
10 ; otherwise, B is equal to 1. 



SEC. II.] 



PLANE TRIGONOMETRY. 



27 



CASE I. 
38. When two angles and a side are given. 

In a plane triangle, ABC, there are given the angle 
A = 58° 07', the angle B = 22° 37', and the side AB, or 
c = 408 yards; to find C, a, and Z». (The sides lying opposite 
the angles A, B, and (7, are denoted by a, h, and c. 

Add the given angles, A and B, to- 
gether, and subtract their snm from 
180° ; the remainder will be the other 
angle, C. Then from the proportion 
(Theorem I), 




Fig. 8. 



sin C : sin ^ : : c : a ; a may be fonnd ; 
and from the proportion, 

sin (7 : sin ^ : : (? : ^ ; h may be found. 



CASE II. 

39. When two sides and an angle opposite one of them are 

given. 

In a plane triangle, ABC, there are given AC, or 5 = 216, 
CB, or a = 117, the angle A = 22° 37', to find the other parts. 

GEOMETRICALLY. 

Draw an indefinite right line A B'B ; 
from any point, as A, draw AC, making 
BAC = 22° 37' and make .4^=216. 
With (7 as a centre, and a radius equal 
to 117, the other given side, describe the 

arc B'B ; draw CB and CB' ; then will either of the triangles, 
ACB or ACB', answer all the conditions of the question. 




28 ELEMENTS OP SURVEYING. [BOOK L 

TRIGONOMETRICALLY. 

From Theorem I, we have, 

a '. 1) : : sin ^ : sin ^, 

By applying logarithms, we have, 

(a. c.) log « (117) 7.931814 

log Z> (216) 2.334454 

log sin ^ (22° 37') .... 9.584968 

log sin B, 45° 13' 55", or 134° 46' 05" . 9. 851236 

The ambiguity in this and similar examples, arises in conse- 
quence of the first proportion being true for either of the angles 
ABC, or AB'C, which are supplements of each other and, there- 
fore, have the same sine (Art. 25). So long as the two triangles 
^(75 and ACB' exist, the ambiguity will continue. But if the 
side CB, opposite the given angle, is greater than AG, the arc 
BB' , described from (7 as a centre and with a radius equal to the 
side a, will cut the line ABB', on the same side of the point A, 
in but one point, and then there will be only one triangle 
answering to the conditions. 

If the side CB is equal to the perpendicular Cd, the arc BB' 
will be tangent to ABB' , and in this case also, there will be but 
one triangle. When CB is less than the perpendicular Cd, the 
arc BB' will not intersect the base ABB' , and in that case no 
triangle can be formed, or it will be impossible to fulfill the con- 
ditions of the problem. 

In the example under consideration, there are two solutions, 
the first corresponding to ^ =: 45° 13' 55", and the second to 
^^'C=: 134° 46' 05". 

FIRST CASE. 

A 22° 3r 

B 45° 13' 55" 

C 180—67° 50' 55" = 112° 09' 05". 



SEC. II.] PLANE TRIGONOMETRY. 29 

Then, in the triangle A CB, 

sin ^ : sin (7 : : b : c, 
and applying logarithms, 

(a. c) log sin ^ (45° 13' 55") .... 0.148764 

log sin C (112° 09' 05") .... 9.9G6700 

log b (216) . 2.334454 

log c 281.785 2.449918 

SECOND CASE. 

A 22° 37' 

B 134° 46' 05" 

C 180° — 157° 23' 05" = 22° 36' 55". 

Then, in the triangle AC'B', 

m\B' : sin C : b : c', 

and applying logarithms, 

(a. c.) log sin B (134° 46' 05") .... 0.148764 

log sin C (22° 36' 55") .... 9.584943 

log b (216) . 2.334454 

log c' 116.993 2.068161 

2. Given two sides of a triangle, 50 and 40 respectively, and 
the angle opposite the latter, equal to 32° ; required the remaining 
parts of the triangle. 

Ans. If the angle opposite the side 50 is acute, it is equal to 
41° 28' 59"; the third angle is then equal to 106° 31' 01", and 
the third side to 72.368. If the angle opposite the side 50 is 
obtuse, it is equal to 138° 31' 01", the thud angle to 9° 28' 59", 
and the remaining side to 12.436. 




Fig. 10. 



30 ELEMENTS OF SUEVEYIN-Q. [bOOK I. 

CASE III. 

40. When two sides and their included angle are given. 

Let ABC he a triangle ; AB and BC, a 

the given sides, and B the given angle. 

Since B is known, we can find the 
sum of the two other angles ; for, 

A + C =1S0''-B, 
and 

i{A^C) =i(lSO°-B). 

We next find half the difference of the angles A and Cy by 
Theorem II, viz., 

BC+BA : BG-BA :: tQXi\{A + C) : tan^(^-C), 

in which we consider BC greater than BA, and therefore A is 
greater than G ; since the greater angle must be opposite the 
greater side. 

Having found half the difference of A and G, by adding it to 
the half sum, |- (^ + C), we obtain the greater angle, and by 
subtracting it from half the sum, we obtain the less. That is, 

l(^A + G)+i{A-G)=A, 
and 

^{A + G)-\(A-G) = a 

Having found the angles A and C, the third side A G may be 
found by the proportion, 

sin ^ : sin ^ : : a : h 



SEC. II.] 



PLAN^E TRIGONOMETRY. 



31 



CASE IV. 

41. Having given the three sides of a plane triangle to find 

the angles. 

Let fall a perpendicular from the angle opposite the greatest 
side, dividing the given triangle into two right-angled triangles ; 
then find the difference of the segments of the base by Theorem 
III. Half this difference being added to half the base, gives the 
greater segment; and, being subtracted from half the base, gives 
the less segment ; the greater segment belongs to the right- 
angled triangle having the greater hypothenuse. We then 
have two sides and the right angle of each of two right-angled 
triangles, to find the acute angles. 

Example. — The sides of a plane 
triangle being given ; viz., BC =^ 40, 
AG =^ 34, and AB = 25 ; required the 
angles. 

BC : AC+AB :: AC-AB : CD-BD. 




That is. 
Then, 

And, 



40 : 59 : : 9 : —^ = 13.275. 



40 + 13.275 

2 

40-1-13.2 75 
2 



40 
= 26.6375 = CD. 



= 13.3625 = BD. 



In the triangle DAC, to find the angle DAC. 
AG : DC :: sin D : sin DAG. 

Applying logarithms, we have, 

(a. c.) log ^(7(34) ....... 8.468521 

log Z>(7 (26.6375) 1.425493 

log sin D (90°) 10.000000 

logsini).4C51°34'40" . , , 9.894014 



'^2 ELKMlCxNTS OF SURVEYING. [BOOK L 

In the triangle BAD, to find the angle BAD. 
,!/>' : /)7> :: sin D : sin BAD. 
^\j)pl}ing logarithms, we have, 

(a. c.) h)gyiy? (25) 8.C020G0 

log BI) (13.302:)) 1.125887 

log sin /> (00°) lO.OQOQOO 

log sin y^y]/) 32° 18' 35" . . . 9.727947 

Henoo, 00^-7)J = 00°— 51° 34' 40" = 38° 25' 20" = C, 

and, [)0'-BAJ) = 00°-32° 18' 35" = 57" 41' 25" == B, 

and, BAD + I)ACz=[)i° 34' 40" + 32° 18' 35" 

= 83° 63' 15" = A. 

42. Solution of Right-angled Triangles. — The nn- 
know n parts of a right-angled triangle maybe found by one of 
tlio last four cases ; or, if two of the sides arc given, by means of 
tlie property that the square of the hypothennse is equal to the 
sum of the squares of the two other sides. Or, the parts may be 
found by Tlieorems IV. and V. 

EXAMPLES. 

1. In a right-angled triangle BACy 
there are given tlie liypotliennse BC = 
250, and the base .ir7 = 240; required ^" 
the other i^nrts. 

A?us. B = 73° 44' 23"; = 1(1° 15' 37"; AB = 70. 

2. In a right-angli'd triangle BAC, there are given, 

J C = 384, and B = 53° 08' ; 
required tlie remaining parts. 

Am. A B =: 287.06 ; BC = 470.070 ; (7 = 3G° 52'. 




FiQ. 18. 



SEC. III.] 



INSTRUMENTS FOE PLOTTIXG. 



33 



SECTION III. 

INSTRUMENTS FOR PLOTTING. 

43. The ordinary implements for making a diagram or plot 
of a survey are— drawing board ; T-square ; dividers ; ruler and 
triangle ; scale of equal parts ; semicircular protractor. 




Pig. 13. 

« 

44. A Drawing Board (Fig. 13) is a rectangular Vjoard of 
about 24 by 30 inches, fths of an inch thick, made of several 
pieces of well seasoned white pine, fitted together with the grain 
running in different directions to prevent warping. It is 
important that its angles should be perfect right angles. 



O 



Pio. 14. 

45. A T-square (Fig. 14) is a ruler about 2 feet in length let 
into a thicker piece of wood at right angles to it. One side of the 
cross-piece is even with the ruler and the other side projects 
somewhat, giving a shoulder on that side. It is a convenient 
instrument for drawing parallels and perpendiculars. 



34 



ELEMEi^TS OF SURVEYIljq^G. 



[book I. 




Fig. 15. 



46. The Dividers (Fig. 15) 
consists of two legs la, he, which 
may be easily turned around a 
joint at b. 

One of the principal uses of 
this instrument is to lay off on a line, a distance equal to a given 
line. 

For example, to lay off on CD, a distance equal to AB. 

Open the points of the dividers to a a| IB 

greater distance than AB, place one point 
lightly upon A, and then slowly and con- 
tinuously close them till the point reaches 
B. Then raise the dividers, place one foot at C, and mark with 
the other the distance CE : this will evidently be equal to AB. 



E 



-D 



Fig. 16. 





Fig. 17. 

47. Ruler and Triangle. — A Ruler of convenient size is 
about twenty inches in length, two inches wide, and a fifth 
of an inch in thickness. It should be made of a hard material, 
perfectly straight and smooth. 

The hypothenuse of the right-angled triangle, which is used in 
connection with it, should be about ten inches in length, and it 
is most conyenient to have one of the sides considerably longer 
than the other. 

The two following problems may be solved with the ruler and 
triangle. 



SEC. III.] INSTKUMENTS FOR PLOTTING. 35 

I. To draw through a given point a line which shall be par- 
allel to a given line. 

48. Let (7 be the given point, and AB the given line. 
Place the hypothenuse of the triangle 
against the edge of the ruler, and then ' — 



place the ruler and triangle on the paper, A B 

60 that one of the sides of the triangle ^^°- ^^• 

shall coincide exactly with AB; the triangle being below the 

line. 

Then, placing the thumb and fingers of the left hand firmly 
on the ruler, slide the triangle, with the other hand, along the 
ruler, until the side which coincided with AB reaches the point 
C. Extend the first and second fingers of the left hand upon 
the triangle to hold it firmly in place, the thumb and remaining 
fingers steadying the ruler, and with the right hand, mark with 
a pen or pencil, a line through C: this line will be parallel 
to AB. 

II. To draw through a given point a line which shall be per- 
pendicular to a given line. 

49. Let AB be the given line, and D the given point. 

Place the hypothenuse of the triangle 
against the edge of the ruler, as before. 
Then place the ruler and triangle so that 
one of the sides of the triangle shall coin- 
cide exactly with the line AB. Then slide 
the triangle along the ruler until the other side reaches the 
point D : then, draw through D, a right line, and it will be 
perpendicular to AB. 

The right angle of the triangle should be carefully tested 
by laying off a perpendicular through the same given point, 



D 
Fig. 19. 



36 



ELEMENTS OF SURYEYIi^^G. 



[book I. 



with the triangle in two positions— to the right of the per- 
pendicular, and then to the left of it. 

50. A Scale of Equal Parts (Fig. 20) is formed by divid- 
ing a line of a given length, into equal portions. 



T..2 .3 .^ .S .C .7 .S .0 10 

I I I I I I I I jzd 



a. 



Fig. 20. 



If, for example, the line ah^ of a given length, say one inch, 
be divided into any number of equal parts, as 10, the scale thus 
formed is called a scale of ten parts to the inch. The line ab, 
which is divided, is called the unit of the scale. This unit is 
laid off several times on the left of the divided line, and the 
points marked 1, 2, 3, &c. 

The unit of scales of equal parts is, in general, either an 
inch, or an exact part of an inch. If, for example, ah, the unit 
of the scale, were half an inch, the scale would be one of 10 parts 
to half an inch, or of 20 parts to the inch. 

If it were required to take from the scale a line equal to 
two inches and six-tenths, place one foot of the dividers at 2, 
on the left, and close the other to .6, which marks the sixth 
of the small divisions : the dividers will then embrace the 
required distance. 



df 



P 







\\ \ \\\\\ 




.09 


\ \ \ \ \ W \ 




.08 


w \ \\y\ \\\\ 




.07 


M M M M M 




■06 M M M 11 Ml 




.0 5 


\ \ \eA \ \ \ 




.0 4 


\ MM M 




.03 


M M M M M 




.02 


M M M M 1 




.01 


1 M M M ( 



2 



1 
9 



a. 12. 3 A .5.6.7 .8 .9 b 



Fig. 21. 



51. A Diagonal Scale of Equal Parts (Fig. 21) is thus 
constructed Take ab for the unit of the scale, which may be 



SEC. III.] INSTRUMENTS FOR PLOTTING. 37 

one inch, J, ^, or f of an incli, in length. On ab describe 
the square abed. Divide the sides ab and dc each into ten equal 
parts. Draw af, and the other nine parallels as in the figure. 

Produce ba, to the left, and lay off the unit of the scale 
any convenient number of times, and mark the points 1, 2, 3, 
&c. Then, divide the line ad into ten equal parts, and through 
the points of diWsion draw parallels to ab, as in the figure. 

Now, the small divisions of the line ab are each one-tenth (.1) 
of ab ; they are therefore . 1 of ad, or . 1 of ag or gh. 

If we consider the triangle adf, we see, that the base df is 
one-tenth of ad, the unit of the scale. Since the distance from 
a to the first horizontal line above ab is one-tenth of the distance 
ad, it follows that the distance measured on that line, between 
ad and af, is one-tenth of df: but since one-tenth of a tenth is 
a hundredth, it follows that this distance is one hundredth (.01) 
of the unit of the scale. A like distance, measured on the second 
line, is two hundredths (-02) of the unit of the scale ; on the 
third, .03; on the fourth, .04, &c. 

If it were required to take, in the dividers, the unit of the 
scale, and any number of tenths, place one foot of the dividers 
at 1, and close the other to that figure between a and b which 
designates the tenths. If two or more units are required, 
the dividers must be placed on a point of division further to 
the left. 

When units, tenths, and hundredths are required, place one 
foot of the dividers where the vertical line through the point 
which designates the units, intersects the line which designates 
the hundredths : then, close the dividers to that line between ad 
and be which designates the tenths: the distance so embraced 
will be the one required. 

For example, to take off the distance 2. 34, we place one foot 
of the dividers at I, and close the other to e : and to take off the 



38 



ELEMENTS OF SURVEYING. 



[book I. 



distance 2.58, we place one foot of the dividers at p and close the 
other to q. 

KoTE 1. — If a line is so long that the whole of it cannot be 
taken from the scale, it must be divided, and the parts of it 
taken from the scale in succession. 



Note 2. — If a line be given upon the paper, its length 
can be found by taking it in the dividers and applying it to 
the scale. 




>' 



Pig. 22. 



62. A Semicircular Protractor (Fig. 22) is used to lay 

down, or protract angles. It may also be used to measure 
angles included between lines, already drawn upon paper. 

It consists of a brass semicircle, ABC. divided to haK 
degrees. The degrees are numbered from to 180, both ways ; 
that is, from A to B and from B to A. The divisions, in the 
figure, are made only to degrees. There is a small notch at 
the middle of the diameter AB, which indicates the centre of 
the protractor. 



SEC. III.] INSTRUMENTS FOR PLOTTING. 39 

To lay off an angle with a Protractor. 

53. Place the diameter AB on the Hne, so that the centre 
shall fall on the angular point. Then count the degrees con- 
tained in the given angle, from A toward B, or from B toward 
A, and mark the extremity of the arc with a pin. Eemove 
the protractor, and draw a line through the point so marked, 
and the angular point : this line will make with the given line 
the required angle. 

The ordinary brass or horn protractors are of but little value. 
Printed protractors of six and twelve inches diameter, upon 
heavy paper, or bristol board, and divided to quarter degrees, 
are very useful and reliable. 

By the following method an angle may be laid off with even 
greater accuracy than with a protractor. With a reliable scale of 
inches, divided to hundredths, set the dividers at five inches, and 
placing one point at the given vertex A (Fig. 
23), describe the arc BO; take out from the y^ 

table the natural sine of half the given angle, y^ 

multiply it by 10, and call the product inches ; ^ b 

with the dividers take off this number of inches ^^^' ^' 

from the scale, and placing one point at B, describe an arc 
cutting BC 2J[> C ; CAB will be the required angle. 

It is evident that twice the sine of half the angle is the chord 
of the whole angle ; and since the radius used is 5 inches, we 
have 

2 (Nat. sine Ja x 5) = 10 x ISTat. sine ^. 



40 



ELEMENTS OF SUEVEYING. 



[book I. 



SECTION IV. 

DEFI N ITIONS. 

54. Surveying comprises all the oj^erartioDS necessary for 
finding the lengths and directions of the bounding lines of any 
portion of the earth's surface, the area of such portion, and for 
making on paper an accurate delineation, or map, of the surface 
and the boundaries. 

55. Plane Surveying is that branch of surveying in which 
the curvature of the earth is neglected, as it may be when the 
survey is limited to small portions of the surface. 

56. G-eodesy, or G-eodetic Surveying, is that branch in 
which the curvature of the earth is taken into account, as it 
must be in all extensive surveys. 

57. A Horizontal Plane at any point is a plane perpen- 
dicular to the radius of the earth drawn to that point. 

58. A Vertical Plane is a plane perpendicular to a hori- 
zontal plane. 

59. A Horizontal Line is any line of a horizontal plane. 

60. A Vertical Line is a line perpendicular to a hori- 
zontal plane. 

61. An Oblique Line is a 

line inclined; i. e., neither par- 
allel nor perpendicular to a hori- 
zontal plane. 

Thus (Fig. 24), AB and DC fig. 24. 

are horizontal lines; BC and 
AB are vertical lines ; and A C and BD are oblique lines. 




SEC. IV.] DEFIKITIOKS. 41 

62. The Horizontal Distance between two points is the 
horizontal line iutercepted between the two vertical lines passing 
through those points. Thus, DC or AB (Fig. 24), is the hori- 
zontal distance between the two points A and C, or between 
the points B and D. 

63. A Horizontal Angle is an angle whose sides are hori- 
zontal ; the plane of its sides is also horizontal. 

64. A Vertical Angle is an angle the plane of whose 
sides is vertical. 

65. An Angle of Elevation is a vertical angle having one 
of its sides horizontal and the other oblique, the oblique side 
being above the horizontal side. Thus, BAC (Fig. 24) is the 
angle of elevation from A to G. 

66. An Angle of Depression is a vertical angle having 
one of its sides horizontal and the other oblique, the oblique side 
being heloiu the horizontal side. Thus, DC A (Fig. 24) is the 
angle of depression from C to A. 

67. An Oblique Angle is an angle the plane of whose 
sides is inclined to a horizontal plane. 



BOOK II. 



CHAIN SURVEYING. 



SECTION I. 

MEASUREMENT OF DISTANCES. 

68. Any tape, rod, or chaiD, divided into equal parts, may be 
Uvsed as a measure for finding the distance between two points. 




Fig. 25. 



The measure in general use for land surveying is a chain of 
four rods, or sixty-six feet in length, called Gunter's chain 
(Fig. 25), from the name of the inventor. It is composed of 100 
links, each joined to the other by two or three rings. Every tenth 
link from either end, is marked by a small attached brass pendant 



SEC. I.] MEASUEEMENT OF DISTANCES. 43 

or tag, which is notched to designate its number from the end. 
The tag at the middle, or fifty-link, point is distinguished by 
being rounded, or by some other peculiarity of make. As the 
tags at equal distances from the two ends of the chain are 
marked the same, care must be taken not to mistake forty links 
for sixty, &c., and the reverse. To avoid such error, it would be 
better to have the tags marked in regular order from the begin- 
ning to the end of the chain, rather than from both ends to the 
middle. 

A link in measure includes a lar, with its connecting ring at 
each end ; when there are three connecting rings, a ring and a 
half at each end is included. 

The handles are of brass, and each forms part of the end link, 
to which it is connected by a nut, by which also the length of 
the chain is adjusted. 

To determine whether to measure from the inside of the brass 
handle, or from the outside, double back the last two or three 
links upon the preceding links and compare. 

The division of the chain into 100 equal parts is very con- 
venient, since the divisions, or links, are decimals of the whole 
chain, and in the calculations are treated as such. 

TABLE. 

1 chain = 4 rods = ^^ feet = 792 inches = 100 links. 
1 link = 7.92 inches. 
80 chains — 320 rods = 5280 feet = 1 mile. 

An excellent chain for accurate measurements is Grum- 
man's patent "suspended chain," which is made of very light 
steel wire, is fitted with spring-balance, thermometer and spirit- 
level attachments, and is held above the surface when in use, the 
ends of the chain being marked upon the ground by the points 
of plummets let fall from the end notches. 



44 



ELEMENTS OF SUBVETING. 



[book II. 



Instead of a chain, which is liable to error because of the 
bars and rings becoming worn by frequent contacts, a steel 
ribbon or tape is often used. 

69. Besides the chain or tape, the surveyor needs ten (or 
better, eleven) marking pins (Fig. 26), made of iron or steel wire, 
about an eighth of an inch in thickness and a foot long, sharpened 
at one end and bent into a ring at 
the other, for marking chain 
lengths on the ground ; a plumbob 
(Fig. 27) and line for referring, 
when necessary, points in the chain 
held horizontally to the inclined 
surface of the ground ; and a set 
of flag-poles, or ranging rods, for 
marking stations and ranging out 
lines. The marking pins should 
be strung upon an iron ring with 
a spring-catch, and this ring 
should be attached to a strap to 
be passed over the right shoulder 
suspending the pins at the left side ; or, better, the pins may be 
carried in a leather quiver strapped to the waist. 

The pins should be tagged with white cloth to enable the 
surveyor to find them again readily, when they have been left to 
mark a point. 

70. To Measure a Horizontal Line. — The point where 
the measurement is to begin is located by a staff temporarily 
placed for the purpose, or by some one of the many permanent 
marks by which the angular points in a boundary are fixed. 

The other extremity of the line must be provided with a staff 
or flag which can be easily seen. 

Two chainmen are required, a fore-chain man, or leader, and 




Fig. 26. 



Fio. 27. 



SEC. I.] 



MEASUREMENT OF DISTANCES. 



45 



a hind-chainman, or follower. The more careful and expert of 
the two should be the follower. 

The leader, with the marking-pins and one handle of the 
chain in his right hand, starts off on the line, drawing out the 
chain to its full length. Both chainmen . now examine it to see 
that there are no inaccuracies in it, either from bent links or 
kinks in the rings joining the links. Having adjusted the chain 
for use, the leader resumes his place, to be directed by the 
follower, who stands behind the staff at the beginning, and sights 
to the staff at the end of the line, so that the measurement shall 
be made exactly along the established line. 

To facilitate this (on level ground) and to insure the correct 
alignment of the pin, at its proper distance, the chain and one 
pin should be held firmly in the 
right hand, as representcid in Figure 
28. While the pin is being aligned, 
it should be held by the leader as far 
from the body as possible, so that the 
view of the flag be left unobstructed. 
To accomplish this, and at the same 
time draw the chain to the proper 
degree of tension, the right arm 
should be braced against the inside 
of the right knee. 

The follower directs, by the simple orders "right" or "left," 
according as the pin, held as described, is to be carried to the right 
or left to bring it into line with the flag. When the pin is truly 
in line, the chain at the same time being drawn straight and 
taut, the order " down " is given, when the leader bringing his 
left hand to bear on the top of the pin, forces it vertically into 
the ground, and resumes his course to the length of another 
chain. 




Fig. 28. 




46 ELEMENTS OF SURVEYING. [BOOK IL 

If, for any reason, the pin can not be 
driven into the ground, the end of the 
chain length should be marked by driving 
the pin olliquelyy always at right angles to 
the chain ; if this cannot be done, a cross 
should be scratched on the ground at the 
exact point, and the pin laid down with its 
point at the mark. 

After one or two chains have been 
measured, on any line, the leader can, by 
glancing back to the station just left, place the pin nearly in' the 
right position ; the exact aligning should be left, however, to 
the follower. 

When the distance to be measured is more than ten chains, 
the pins, when exhausted, should be returned to the leader, the 
distance noted in a field-book provided for the purpose, and the 
chaining recommenced at the place of the tenth pin. If but ten 
pins are used, the follower has then but the hole made by the 
tenth pin to measure from. For this reason, eleven pins are 
often used, the eleventh being of different material from the 
others for distinction, and is used by the follower to measure 
from when the ten pins are returned to the leader. 

71. All distances should be measured horizontally. Hence, 
when the ground slopes, one end of the chain must be elevated. 
Each chainman should be provided with a small plumb-line, so 
that the elevated end of the chain may be held directly over the 
proper point. 

. "When the raised end of the chain is only two, or even three 
feet above the ground, it will suffice, in many cases, to use a 
marking-pin, held lightly by the point, between the thumb and 
finger, instead of a plumb-line. When the chaining is on a steep 
inclination, other precautions should be observed. 



SEC. I.] 



MEASUREMENT OF DISTAKCES. 



47 



Suppose the chaining to be up hill. The leader draws the 
chain out to its full lengthj as in any other case, and then returns 




Fio. 30 

to within such a distance of the follower, that when the chain is 
drawn out to that length horizontally, it shall not be too high to 
be held conyeniently. 

The follower holds his end of the chain carefully over the 
point or station, by means of the plumb-line, while he directs the 
leader in the usual manner. 

The point fixed in this manner, by the leader, must not be 
marked by a marking-pin, but by a small peg or nail. At the 
order, "Down," the leader does not go forward immediately, but 
waits until the follower comes up and takes the chain by the 
precise point held, the moment before, to the ground. 

This point is now held above the peg by the the follower, 
who uses the plumb, as before, and aligns the leader, who has 
taken hold of the chain a few links farther on, and is holding it 
to the ground. These short distances are not recorded. The end 
of a full chain is marked by a marking- pin. 



48 



ELEMENTS OF SURVEYIN"G. 



[book II. 



In chaining doivn-hill, the method is essentially the same. 
The leader uses the plumb, and determines by it where the peg 
is to be placed. 

At the end of a course, the part of a chain is measured by 
drawing the chain onli/ to the flag, where it is held by the leader, 
until the follower comes forward to the last pin, and counts 
the links. 

In measuring up the hill from 
A to C, or down the hill from 
C to A, the horizontal distances 
ahf c d, and / C, are measured and 
their sum is the horizontal dis- 
tance between A and C. 

Chaining doion hill gives more 
accurate results than chaining up ; for in chaining down, the 
follower holds the chain firmly upon the ground and no ordinary 
pull by the leader moves it. It is impossible to hold a chain 
perfectly steady over a point, by means of a long plumb-line, 
while the leader is pulling out. 




Fig. 31. 



72. When the ends of a line can not be seen, each from the 
other, intermediate points between the two must be established. 
When a hill intervenes, such points may be established thus : let 
the surveyor and an assistant, each with a ranging-rod, place 
themselves as nearly in the line as possible, and in such position 
that each can see the other and the flag beyond him. The sur- 
veyor looking to the flag at the end of the line, directs the 
assistant into line with it ; the assistant then looks to the flag 
at the beginning of the line and directs the surveyor into 
line with it ; the surveyor from his new position redirects 
the assistant into line with the end flag-stafl ; the assistant 
then realigns the surveyor with the flag-staff at the beginning 
of the line ; the operation is repeated till both stand in the 



SEC. I.J 



MEASURExMENT OF DISTANCES. 



49 



desired Hue, when their positions are marked with the rang- 
ing-rods. 

73. When a valley is to be chained across, intermediate 
points in the lower portions of it may be fixed, if necessary, by 
the surveyor holding a plumb-line so as to cover the flag-staffs at 
both ends of the line and directing an assistant to fix, between 
the two, ranging-rods which shall also be covered by the plumb- 
line. 

74. When a wood intervenes between the two ends of a line, 
a trial line may be run out by ranging-rods placed at convenient 
distances in line with each other and with the staff at the 
beginning of the line, and as nearly as possible in the required 
line. Then draw on the ground a perpendicular (by a method 
to be shown presently) from the staff at the end of the required 
line to the trial line just run out, and measure the length of 
this perpendicular. The ranging-rods may then, by the property 
of similar triangles, be put in their true position in line. Thus, 




^ G^ is to be measured ; the trial Ime AF \^ run out and the 
ranging-rods B, C, &c., fixed at known distances apart and as 
nearly in the line as possible ; the perpendicular 6^i^is measured ; 
then from the similar triangles, AFG and ABH, 

AF: AB : : FG : BH, 

The distance BH thus becomes known, and the ranging-rod 
at B is moved, on a perpendicular to AF, the required distance 
to its true position at H. The other ranging-rods are in like 



50 



ELEMENTS OF SURVEYING. 



[book II. 




Fig. 33. 



manner put in their true positions at /, K, &c., and the true line 
is marked out. 

75. To trace on the ground the direction of a straight line, 
that shall be perpendicular, at a given point, to a given 
straight line. 

FIRST METHOD. 

Any three lines having the ratio 3, 4, and 5, form a right- 
angled triangle. 

Let AB (Fig. 33) be 
the given line and C the 
point at which the per- 
pendicular is to be drawn. 
Divide the number of 
links in the chain by 8, 
neglecting the remainder ; 
if we are using the 100-link chain the quotient would be 12 links. 
From the point C measure a distance towards A equal to four 
times this quotient (48 links) ; place one end of the chain at (7, 
and the end of the 96th link [(5 + 3) x 12 = 96] at A', then 
taking the end of the 36th link (3 x 12) pull out the chain so 
that the two portions EA and EG are taut, and E will be a 
point on the perpendicular required. 

This method supposes the chain to be correctly divided into 
links. 

SECOND METHOD. 

Let AD be the given right line, 
and D the point at which the perpen- 
dicular is to be drawn. Take the 
longest available distance on the tape 
or chain (the whole of it if possible), 
and place one extremity at D, and 
fasten the other at some point, as E, 




Fig. 34. 



SEC. I.] 



MEASUREMENT OF DISTANCES. 



51 



between the two lines which are to form the right angle. Place 
a staff at E. Then, having stationed a person at D, remove that 
extremity of the chain and carry it round until it ranges on 
the line DA, at A. Place a staff at A : then remove the end 
of the chain at A, and carry it round until it falls on the line 
AB, prolonged, at F. Then place a staff at F; ADF will be 
a right angle, being an angle in a semicircle. 

This method is independent of any errors of graduation of 
the chain ; it also gives the largest possible construction in the 
field, a matter of importance as insuring the most correct 
results. 

76. There is a simple instrument for laying off right angles on 
the ground called the Surveyor's 
Cross. This instrument con- 
sists of two bars, AB and (72), 
Fig. 35, permanently fixed at 
right angles to each other, and 
firmly attached at F, to a 
pointed staff, which serves as 
a support. Four sights are 
screwed firmly to the bars, by 
means of the screws a, b, c, 
and d. 

As the only use of this in- 
strument is to lay off right 
angles, it is of the first im- 
portance that the lines of sight 

be truly at right angles. To ascertain if they are so, let the bar 
AB be turned until its sights mark some distinct object ; then 
look through the other sights, and place a staff on the line which 
they indicate ; let the cross be then turned until the sights 
of the bar AB come to this last line; if the other sights are 




Fio. 35. 



5^ ELEMENTS OF SURVEYING. [bOOK II, 

directed to the first object, the lines of sight are exactly at 
right angles. 

The sights being at right angles, if one of them be turned 
in the direction of a given line, the other will mark the direc- 
tion of a line perpendicular to it, at the point where the instru- 
ment is placed. 

77. From a given point without a straight line, to let fall a 
perpendicular on the line. — Let C be the given point, and AB 
the given line (Fig. 36). 

From C, measure a line, as CA, . 

to any point of the line AB. From o JSf'^ 

A, measure on AB any distance as i,,^'^ \ 
AF, and at F erect FE perpendicu- 

Fio. 36. 

lar to AB. 

Having stationed a person at A, measure along the perpen- 
dicular FE until the forward staff is aligned on the line AC: 
then measure the distance AE. From similar triangles, 

AE: AF :: AC: AD; 

in which all the terms are known except AD, which may, there- 
fore, be found. The distance AD being laid off from A, the 
point Df at which the perpendicular CD meets AB, becomes 
known. If the length of the perpendicular is desired, it may be 
found from the proportion, 

AE: EF :: AC: CD, 

in which all the terms are known except CD. 

It is always best to make AF nearly equal to AD when 

practicable ; for then AE will be nearly equal to AC, and the 

multiplication of errors will be avoided ; in the expression 

AC AC 

AD = -j^xAF, -j-p will be but little greater than 1. 



SEC. I.] 



MEASUREMENT OF DISTANCES. 



53 




Fig. 37. 



78. To trace on the ground a straight line that shall pass 
through a given point and be parallel to a given straight line. 

Let ABhe the given 
line and P the given 
point. From F meas- 
ure any oblique line to 
AB, as PQf and mark 
its middle point, which 
call m in the figure. From any point of AB, as E, run a line 
through m, and prolong it till mS = Rm ; then PS will be the 
parallel required. 

79. To determine the horizontal distance from a given point 
to an inaccessible object. 

Let A be an inaccessible object, 
and B the point from which the dis- 
tance is to be measured. 

FiEST Method. — At the point B, 
lay off BB perpendicular to the line 
BA, and measure along it any con- 
venient distance, as BB. 

At B lay off the right angle EBD, and measure any distance 
in the direction BD, Let a person at D align a staff on DA, 
while a second person at B aligns it on BB -. the staff will thus 
be fixed at C. Then measure the distance BC. 

The two triangles BOD and CAB being similar, we have, 

BG : BD \: CB : BA, 

in which all the terms are known, except the fourth, which is, 
therefore, found. 

11 BC can be made equal to CB, then the measured distance 
BD is the distance required. The triangle BDC should be as 



1^ 



B 



D 



Fig 3S. 



54 



ELEMENTS OF SUETEYING. 



[book II. 




nearly equal to ACE as practicable, and the angle BCD should 
not exceed 45°. 

Secoxd Method.— Let B be the given point, and A the 
inaccessible object ; it is required to find BA. 

Measure any horizontal base-line, 
as BC. Then, having placed staves 
at B and C, measure any conre- 
nient distances BD and CE, such 
that the points D, B, and A^ shall 
be in the same right line, as also, 
the points E, C, and A : then meas- 
ure the diagonal lines DC and EB. 

Kow, in the triangle BEC, the fig. 39. 

three sides are known, therefore, the angle ECB can be found. 
In the triangle CDB, the three sides are also known, therefore, 
the angle CBD can be determined. These angles being respec- 
tively subtracted from 180°, the two angles ACB and ABC 
become known; and hence, in the triangle ABC, we have two 
angles and the included side, to find the side BA. 

The lines BD and CE should each equal BC,if. possible, to 
facilitate computation, and the angle at A should not be less 
than 10°. 

n 

Thied Method. — Let AC he the 

distance required. Lay off the right 

angle CAB, and measure AB, any 

convenient distance. At B lay off the 

right angle CBD, and fix the point D, 

carefully, in line with AC. Measure 

AD. Then, 

AB^ 



AD : AB :: AB : AC..-.AC = 



AD 



(Legendre, Bk. IV. Prop. 23). 

KoTE. — When such problems occur 




Fig. 40. 



SEC. I.] 



MEASUREMENT OF DISTANCES. 



55 



in practice, the distance AC \s usually a portion of a longer line, 
so that the line CAD is well marked by stakes or pins, before AB 
is measured. 

80. To prolong a line beyond an obstacle. — If the obstacle 
can be seen over, the surveyor should send an assistant, 
with a flag-pole or ranging-rod, to the further side of it, to 
a point approximately in line. At this point the assistant 
should hold the rod vertical and exposed to the surveyor, at one 
end of the line, who directs him to ^' right" or "left" till the 
ranging-rod covers or coincides with the flag-pole at the further 
end of the line, when it is inserted in the ground and marks a 
point in the desired line. Other points may be determined in 
like manner. This is called "ranging," or "ranging out" the 
line. 

When the obstacle cannot be seen over, the continuation may 
be effected as follows : 

First Method. — Let OA 
be the line to be prolonged. 
Lay off OAB = 120°, or 
CAB = 60". Measure AB, 
of such length as to permit 
BC to be measured without 
meeting the obstruction. 
Make ABC = 60°, and 
measure BC, equal to AB. 
If ^ be not in sight from C, 

make the angle BCP equal to 120°, and resume the survey of 
the line. AC is equal to AB or BC. 

Note. — This method may be employed in the absence of any 
angular instruments, by constructing an equilateral triangle with 
the chain. Measure half a chain from A towards C; then fasten 








56 



ELEMEISTS OF SURVEYIH^G. 



[book II. 




Fig. 42. 



the ends of the chain at A and the point so determined, and pull 
out the middle, thus forming an equilateral triangle. If there is 
not room to measure towards C, measure back towards 0, and 
construct the angle on the side of A away from B. The errors 
are cumulative in this method, but it is rapid, and will do for 
work not requiring great accuracy. 

Second Method. 
Take two points, A 
and B, at conyenient 
distance apart, one or 
two chains, and draw 
two offsets, AB and 

BF, at right angles to the direction of the line and of sufficient 
length to clear the obstacle ; draw^ BF through the extremities of 
the rectangular offsets and prolong it beyond the obstacle ; this 
last line ^vill be parallel to the original line of direction ; at G 
and H draw the lines at right angles to Bff, and make them 
equal in length to AB and BF; draw CD, it will be the pro- 
longation required. FG is equal in length to BC, provided the 
perpendiculars BF and GO are accurately laid out. This is a 
good method to use in passing trees or small obstacles, provided 
the distance between perpendiculars AB and BF, also CG and 
DH, is one chain or more. The perpendiculars are often laid off 
by- guess, which method will give a very fair prolongation, but 
the distance BC thus obtained will not be accurate. 




Third Method. — From any point A, measure AB; through 
its middle point x, run CD, making Dx = Cx ; then DB will be 



SEC. I.] 



MEASUREMEN^T OF DISTAI^CES. 



57 



parallel to A C. From D run a line, DE, through any point of 
the line AB, as y, making yE of such length that, 

By : Dy :: Ay : Ey. 

Then through z, the middle point of DE, run BH, making 
zH = Bz. HE will be the prolongation of A C. 
To find CH we have. 

By : BD : : Ay : AE, 

fi'om which AE is known ; 

AE—(AG-\^ HE) = CH, 

81. To find the altitude of an object, when the distance to 
the vertical line passing through the top of it is known. — Let 
CD be the altitude required, and A C the known distance. 

From A, measure on 
the line A C, any conveni- 
ent distance AB, and place 
a staff vertically at B, 
Then placing the eye at A, 
sight to the object D, and 
let the point, at which the 
line AD cuts the staff BE, be marked. Measure the distance 
BE on the staff ; then, 

AB : BE :: AC: CD, 

whence CD becomes known. 

If the line ^(7 cannot be measured, on account of intervening 
objects, it may be determined by calculation, as in the preceding 
article, and then, having found the horizontal distance, the 
vertical line is readily determined, as before. 

82. Standard. — As the chain varies in length from changes 
of temperature and from use, it should be daily compared with 




Fig. 44. 



58 ELEMENTS OF SURVETIXG. [BOOK IL 

a standard kept for the purpose. A convenient standard for 
such comparison may be made by driving into a level and even 
piece of ground two stakes, sawed off even with the surface of 
the ground, distant from each other one chain, or CG feet, 
accurately measured, with nails driven into the heads of the 
stakes to mark the exact length of the standard. Marks made 
upon the coping of a wall, or a curb-stone, will answer the same 
purpose. 

If it is found that any line has been measured with either too 
long or too short a chain, the true distance may be found by the 
proportion ; 

The length of the standard 
: the length of the incorrect chain used 
: : the measured distance 
: the true distance. 

For areas the proportion would be ; 

The square of the length of the standard 
: the square of the length of the chain used 
:: the area found 
: the true area. 



SECTION II. 

AREA OR CONTENTS OF GROUND. 

83. The surface of ground being, in general, broken and 
uneven, it is impossible, without great trouble and expense, 
to ascertain its exact area or contents. To avoid this incon- 
venience, it has been agreed to refer every surface to a hori- 
zontal plane; that is, to regard all its bouHding lines as hori- 



SBC. ll.j 



AREA OR CONTENTS OF GROUND. 



59 



^A: % 




Fig. 45. 



zontal, and its area as measured by that portion of the 
horizontal plane which the boundary lines enclose. 

For example, if ABCD were a 
piece of ground, having an uneven 
surface, the whole would be referred 
to a horizontal plane, and that part 
of the plane which is included be- 
tween the bounding horizontal lines 
AB, BC, CD, DA, be taken for the 
measure of the area. 

In estimating land in this manner, the sum of the areas of 
all the parts, into which a tract may be divided, is equal to the 
area, estimating it as an entire piece ; but this would not be 
the case if the areas of the parts had reference to the actual 
surface, and the area of the whole were calculated from its 
bounding lines. 

84. The unit of measure of any quantity is a quantity of 
the same kind, regarded as a standard. For lines, the unit is a 
right line of a known length, as 1 foot, 1 link, 1 chain, or any 
other fixed distance. In measuring land, the length of Gunter's 
chain is generally taken as the unit of linear measure. 

85. The unit of measure for surfaces is a square described 
on the unit of linear measure. 

When, therefore, the linear measures are feet, yards, rods, or 
chains, the superficial measures, are square feet, square yards, 
square rods, or square chains ; and the numerical expression for 
the area, is the number of times which the unit of superficial 
measure is contained in the land measured. 

An Acre, which is the common unit of measure for land, is 
a surface equal in extent to 10 square chains ; that is, equal to a 
rectangle of which one side is ten chains and the other side one 
chain. 



60 



ELEMENTS OF SURVEYI2S"G. 



[book II. 



A Rood, is one quarter of an acre. 

Since the chain is four rods in length, 1 square chain con- 
tains 16 square rods ; and therefore, an acre, which is 10 square 
chains, contains 160 square rods, and a rood contains 40 square 
rods. A square rod is called a perch. 

86. Land is generally computed in acres, roods, and perches, 
which are respectively designated by the letters A. R. P. 

When the linear dimensions of a survey are chains or links, 
the area will be expressed in square chains or square links, and it 
is necessary to form a rule for reducing such area to acres, roods, 
and perches. The reduction may be made by the following 









TABLE. 




Miles. 


Acres. 


Roods. 


Sq. Chains. 


Perches. 


Sq. Links. 


1 


640 
1 


2560 
4 
1 


6400.0 

10.0 

2.5 

1.0 


102,400 

16a 

40 

16 

1 


64,000,000 

100,000 

25,000 

10,000 

625 



1 square mile = 6400 square chains = 640 acres. 

When the linear dimensions are links, the area will be ex- 
pressed in square links, and may be reduced to acres by dividing 
by 100,000, the number of square links in an acre ; that is, by 
pointing off five decimal places from the right hand. 

If the decimal part be then multiplied by 4, and five places of 
decimals pointed off, in the product, from the right hand, the 
figures to the left will express the roods. 

If the decimal part of this result be now multiplied by 40, 
and five places for decimals pointed off, as before, the figures to 
the left will express the perches. 

If one of the dimensions be in links, and the other in chains, 



sec: II.] AREA OR CONTENTS OF GROUND. 61 

the chains may be reduced to links by annexing two ciphers ; or, 
the multiplication may be made without annexing the ciphers, 
and the product reduced to acres and decimals of an acre, by 
pointing off three decimal places from the right hand. 

When both the dimensions are in chains, the product is 
reduced to acres by dividing by 10, or pointing off one decimal 
place. 

From which it is evident that, 

1st. If links he multiplied hy linhs, the product is reduced to 
acres by pointing off five decimal places from the right hand. 

2d. If chains he multiplied hy links, the product is reduced to 
acres hy pointing off three decimal places from the right hand, 

3d. If chains he multiplied hy chains, the product is reduced 
to acres hy pointing off one decimal place from the right hand, 

87. Since there are 16.5 feet in a rod, a square rod is equal 
to 

16.5 X 16.5 = 272.25 square feet. 

If the last number be multiplied by 160, there will result 

272.25 X 160 = 43560 = the square feet in an acre. 

Since there are 9 square feet in a square yard, the last number 
divided by 9, will give 

4840 = the number of square yards in an acre. 

88. To find the area of a piece of ground in the form of ;\ 
square, rectangle, or parallelogram. 

Rule. — Multi-ply the base by the altitude, and the 
product will express the area (Geom., Bk. IV., Prop. IV.). 



62 



ELEMENTS OF SUBVEYING. 



[book IL 



Example. — To find the area of the rec- 
tangular field A BCD. 

Measure the two sides AB, BC ; suppose 
that AB = 14 chains 27 links, and BC=d 
chains 75 links. Then, 




Fig. 46. 



AB = 


1427 links. 


BC = 


975 links. 


ABxBC = 


1391325 square links. 


= 


13.91325 acres. 




4 

3.65300 roods. 




40 




26.12000 perches. 



Ans. 13 A. S R, 26 P. 



89. To find the contents of a piece of land in the form of a 

triangle. 

FIRST METHOD. 

KuLE. — Measure either side of the 
triangle as BC, and from the opposite 
angle A, let fall a perpendicular AD, 
and measure this per^pendicular ; 
then, multiply the base and perpen- 
dicular together, and divide the 

product by 2 ; the result mill express the area of the 
triangle (Geom., Bk. IV., Prop. VI.). 




EXAMPLES. 



1. What are the contents of a triangle whose base is 25 ch. 
1 1., and perpendicular 18 ch. 14 1. ? Ans, 22 A. 2 R. 29 P. 



SEC. IL] AKEA or COl^TENTS OF GROUND. 63 

2. What are ttie contents of a triangle whose base is 15.48 
chains, and altitude 9.67 chains? Ans, K A. 1 E. 38 P. 

SBCONDMETHOD. 

Rule. — Measure the three sides of the triangle, Then^ 
add them together and tahe half their sum. From this 
half sum subtract each side separately. Then, multiply 
the half sum and the three remainders together, and 
extract the square root of the product ; the result will he 
the area (Geom. Mens., Art. 97). 

Or, after having obtained the three remainders, add 
together the logarithm of the half sum and the logarithm's 
of the respective remainders, and divide their sum by 2 ; 
the quotient will be the logarithm of the area. 

EXAMPLES. 

1. Find the area of a triangular piece of ground whose sides 
are 20, 30, and 40 chains. 

BY FIRST BULE. 

20 45 45 45 
30 -20 —30 —40 
40 25 1st rem. 15 2d rem. 5 3d rem. 
2)90 

45 = half sum. 

Then, 45 x 25 x 15 x 5 = 84375 ; 

and V^SSSTS = 290.4737 = the area. 

Ans. 29 A. R. 8 P. 

2. What is the area of a triangle whose sides are 2569, 4900, 
and 5035 links? 



H ELEMENTS OF SURVEYING. [BOOK II. 

BY SECOND RULE. 

2569 6252 6252 6252 

4900 — 2569 —4900 —5035 

5035 3683 Ist rem. 1352 2d rem. 1217 3d rem. 



2 ) 12504 



6252 = half sum. 



Then, log 6252 3.796019 

log 3683 3.566202 

log 1352 3.130977 

log 1217 3.085291 

2 )13.578489 

Area in square links, 6155225 . . . 6.789244 

Am. 61 A. 2 E. 8 P. 

90. To find the area of a piece of land in the form of a 

trapezoid. 

Rule. — Measure the two parallel sides, and also the per- 
pendicular distance between them. Add the two parallel 
sides together, and take half the sum ; then multiply the 
half sum hy the perpendicular, and the product will be the 
area (Geom., Bk. IV., Prop. VII.). 

EXAM PLES. 

1. What is the area of a trapezoid, of 
which the parallel sides are 30 and 49 chains, 
and the perpendicular distance between them 



16 eh. 60 1., or 16.60 chains ? Fig. is. 



SEC. II,] 



AEEA OR CONTENTS OF GROUND. 



65 



30 + 49 = 79 ; dividiDg by 2, gives . . . 39.5 
Multiply by .16.60 



Area in square chains 655. 700 

Ans. 65 A. 2 R. 11 P. 

2. Eequired the contents, when the parallel sides are 20 and 
32 ch., and the perpendicular distance between them 26 ch. 

Ans. 67 A. 2 R. 16 P. 



91. To find the area of a piece of land in the form of a 

quadrilateral. 

Rule. — Measure the four sides of the quadrilateral, 
and also one of the diagonals ; the quadrilateral will thus 
he divided into two triangles, in both of ivhich all the sides 
will he hnown. Then, find the areas of the triangles 
separately, and their sum will he the area of the quad- 
rilateral. 

Example. — Suppose that the sides 
and diagonal AC, of the quadrilateral 
ABCD have been found, 

AB z=z 40.05 ch., CD = 29.87 ch., 
BC = 26.27 ch., AD = 37.07 ch.j 
and ^C = 55 ch.; 

required the area of the quadrilateral. 

Ans, 101 A. 1 B. 15 P. 

Note. — Instead of measuring the four sides of the quadri- 
lateral, the perpendiculars Bb, Dg, may be let fall on the 
diagonal AC The area of the triangle may then be determined 
by measuring these perpendiculars and the diagonal AC, The 
perpendiculars are Dg = 18.95 ch., and Bh = 17.92 ch. 




66 



ELEMENTS OF SURVEYING. 



[book II, 



92. To find the contents of a field having any number of 

sides. 

Rule. — Measure the sides of the field and also the 
diagonals ; the three sides of each of the triangles into 
which the field mill he thus divided will then he known, 
and the areas of the triangles may then he calculated hy 
the preceding rules. Or, measure the diagonals, and from 
the angular points of the field draw perpendiculars to the 
diagonals and measure their lengths ; the hase and per- 
pendicular of each of the triangles will then he known. 

Example. — Let it be required to determine the contents 
of the field ABODE, having five sides. 

Suppose that the diagonals and per- 
pendiculars have been found, 

^(7=36.21 ch., ^6'r= 39.11 ch., 

Bh=: 4.08 ch., Dd.zzz 7.26 ch., 

^a = 4.19ch.; 

required the area of the field. 




Fig. 50. 



Area of triangle ABO = 73.8684 square chains, 
Area of " CZ>^= 141.9693 " 
Area of " AOE = 81.7399 " 
Area of " ABODE = 297.5776 '^ 

Ans, 29 A.3E,1 F. 



93. To determine the area, when the diagonals and per- 
pendiculars cannot be measured; as in the case of swamp or 
submerged meadow. 

TiULE.— Measure the hounding lines of the area, and 
then determine the diagonals hy outside tie-lines. 



SEC. II.] 



AREA OK COl^TENTS OF GROUND. 



67 



Thus, let ^^CZ)J5' repre- 
sent the polygon including 
the area, and suppose its 
sides to be measured. Pro- 
long BA to X, making Ax 
any exact part of AB — say 
one third — and also prolong 
FA to y, Ay being one 
third of AE; then, because 

of similar triangles, the measured length of xy will be one third 
of BK In like manner CB may be determined. Having the 
perimeter and the diagonals proceed as in Art. 92. 




II 



94. To find the contents when the boundary is an irregular 

line. 

It frequently happens that 
a plot to be surveyed is 
bounded partly by an irregu- 
lar line. In such a case, 
one or more straight lines 
are surveyed, and offsets 
measured from these lines, 
as often as may be required 
to afford data for the compu- 
tation of the area and a true 

delineation of the boundary. In the case represented in the 
figure, the stream from A to C is the boundary. The station 
B is selected for convenience, as it is evident, if the line were 
run direct from A to C, the labor of taking the offsets would be 
much greater. 

It will be observed that the offsets are so measured as to 
indicate the abrupt bends in the boundary; and furthermore, so 




68 



ELEMEIJ^TS OF SURVEYING. 



[book II. 



that the areas thus cut off may be considered as Veing bounded 
by straight lines, without sensible error. 

When the boundary is a 
crooked stream that is easily 
crossed, it is often convenient 
to survey a line across the 
bend, as in the figure, and 
locate by offsets upon both 
sides of the line. In any 
case, the small areas to be 
computed are only trapezoids 
and triangles. 




Fig. 53. 



95. To find the area of a piece of ground in the form of s 

circle. 



Rule. — Measure the radius AG; 
then multiply the square of the radius 
by 3.I4I6 (Mens., Art. 105). 




Fig. 54. 



To find the area of a circular piece of land, of which the 
diameter is 25 ch. Ans. 49 A. R. 14 P. 



96. To find the contents of a piece of ground in the form of 

an ellipse. 



Rule. — Measure the semi-axes AE, 
CE. Then multi-ply them together, and ^ 
their product by 3.I4I6, 




SEC. II. 1 AREA OR COKTEKTS OF GROUND. S19 

To jSnd the area of an elliptical piece of ground, of which 
the transverse axis is 16.08 ch., and the conjugate axis 9.72 ch, 

Ans, 12 A.1R4: F, 

Note 1. — The following is the manner of tracing an ellipse 
on the ground, when the two axes are known. 

From Cy one of the extremities of the conjugate axis as a 
centre, and AU, half the transverse axis, as a radius, describe 
the arc of a circle cutting AJS in the two points F and G ; 
these poins are called the foci of the ellipse. 

Then, take a tape, the length of which is equal to AB, 
and fasten the two ends, one at the focus F, the other at the 
focus G. Place a pin against the tape and move it around, 
keeping the tape tightly stretched : the extremity of the pin 
will trace the curve of the ellipse. 

Note 2. — In determining the contents of ground, in the 
examples which have been given, the linear dimensions have 
been taken in chains and decimals of a chain. 

If the linear dimensions were taken in terms of any other 
unit, they may be readily reduced to chains. For, a chain is 
equal to 4 rods, equal to 22 yards, equal to 66 feet. Hence, 

1st. Rods may be reduced to chains and the decimal of a 
chain, ly dividing hy 4. 

2d. Yards may be reduced to chains and the decimal of a 
chain, by dividing by 22, 

3d. Feet may be reduced to chains and the decimal of a 
chain, by dividing by 66. 

Note 3. — If it is thought best to calculate the area, with- 
out reducing the linear dimensions to chains, the result can 
be reduced to acres* 



70 |:lements of surveyixg. [book ii. 

1st. By dividing it by 160, when it is in square rods 
(Art. 85). 

2d. By dividing it by J^Jfi, when it is in square yards 
(Art. 87). 

3d. By dividing it by 43560, when it is in square feet 
(Art. 87.) 



BOOK III. 

COMPASS SURVEYING. 



SECTION I. 

DEFIN ITIONS. 

97. The Axis of the earth is the immovable diameter about 
which it revolves ; and the poles are the points in which the 
axis meets the surface. 

98. Any plane passing through the axis of the earth is called 
a meridian plane ; and its intersection with the surface is called 
a meridian line, or simply a meridian. 

99. All the meridians converge towards the poles, but they 
vary so little from parallelism, within the narrow limits of 
surveys made with the compass, that they may, without sensible 
error, be regarded as parallel straight lines. 

100. K a magnetic needle be suspended freely, and allowed 
to settle to a state of rest, a vertical plane passed through its 
axis is called the plane of the magnetic meridian ; and its inter- 
section with the surface of the earth is called the magnetic 
meridian, or sometimes, a North and South line. A line per- 
pendicular to a North and South line, is called an East and 
West line. 

101. A line traced, or measured on the ground, is called a 
Course; and the angle which this line makes with the magnetic 




72 ELEMENTS OF SURVETIKG. [BOOK III 

meridian, passing through the point of beginning, is called the 
Bearing. 

Thus, if we start from the point 
A, and measure in the direction AB, 
the line AB is the course, and the 
angle NAB is the bearing. 

When the course, like AB, falls 
between the north and east points, 
and makes an angle of 46° with the 
meridian, the bearing is read, north FioTse. 

46° east, and is written, K 46° E. 

When the course, like ACy falls between the north and west 
points, and makes with the meridian an angle of 30°, the bearing 
is read, north 30° west, and is written, N. 30° W. 

When the course, like AD, falls between the south and west 
points, and makes an angle with the meridian of 70°, the bearing 
is read, south 70° w^est, and is written, S. 70° W. 

When the course, like AF, falls between the south and east 
points, and makes with the meridian an angle of 70°, the bearing 
is read, south 70° east, and is written, S. 70° E. 

A course which runs due north, or due south, is designated 
by the letter N, or S ; and one which runs due east, or due west, 
by the letter E, or W. 

102. If, after having passed over a course, the bearing is 
taken to the back station, this bearing is called the lack sight, or 
reverse hearing. 

103. The perpendicular distance between the east and west 
lines, drawn through the extremities of a course is called the 
northing or southing, according as the course is run towards the 
north or south. This distance is also called the difference of 
latitude, or simply the latitude, because it shows the distance 
which one end of the course is north or south of the other. 



SEC. I.J DEFINITIONS. 73 

Thus, in running the course from A -^ 



to B, AC is the difference of latitude, c 

north. 



^^ 



-re 



E 



S 

Fig. 57. 



w- 

104. The perpendicular distance between 
the meridians passing through the extremi- 
ties of a course, is called the departure of 
that course, and is east or west, according 

as the course lies on the east or west side of the meridian passing 
through the point of beginning. 

Thus, in running the course AB, CB is the departure, east. 

105. It is found convenient, in explaining the rules for sur- 
veying with the compass, to attribute to the latitudes and 
departures the algebraic signs, -|- and — . 

We shall, therefore, consider every northing as affected with 
the sign +, and every southing as affected with the sign — . 
We shall also consider every easting as affected with the sign +, 
and every westing as affected with the sign — . 

106. The meridian distance of a point is its perpendicular 
distance from any assumed meridian. Thus, if the distance be 
estimated from the meridian NS, BC will be the meridian dis- 
tance of the point B. 

107. The meridian distance of a line, is the meridian dis- 
tance of its middle point, and is east or west, according as this 
point lies on the east or west side of the assumed meridian. 
Thus, FG drawn through the middle point of AB, is the 
meridian distance of the line AB. 

The sign -f- will always be given to the meridian distance 
of a point or line, when it lies on the east of the assumed 
meridian, and the sign — , when it lies on the west. 



74 



ELEMENTS OF SURVEYIIi^G. 



[book III. 



SECTION II. 

SURVEYOR'S COMPASS. 

108. The Surveyor's Compass consists of a compass-box, 
DCE\ a magnetic needle; a brass plate, AB, from twelve to 
fourteen inches long; two plana sights, ^i^and BG ; two spirit- 




FiG. 58. 



levels placed at right angles to each other ; a brass head, K, to fit 
the compass to a stand, which is sometimes a tripod and some- 
times a single staff, called Jacob-staff, pointed with iron at the 
lower end so that it may be placed firmly in the ground. 

109. The compass-box DCE is circular, and generally about 
six inches in diameter. At the centre is a small pin, on which 
the magnetic needle is poised. This needle, if allowed to turn 
freely around the point of support, will settle to a state of rest ; 



SEC. II. J SUE VETO R'S COMPASS. 75 

the direction which it then indicates, is that of the magnetic 
meridian. 

In the interior of the compass-box, there is a graduated 
circle divided to degrees, and half degrees ; the degrees are 
numbered from the extremities of the diameter NS, both ways 
to 90°. 

The length of the magnetic needle is a little less than the 
diameter of the graduated circle, so that the needle can move 
freely around its centre, within the circle, and its positions be 
noted on the graduated arc. 

110. The open sights, AF and BG, are placed at right 
angles to the plate AB, and fastened to it firmly by screws. 
The sights have fine slits cut through nearly their whole 
length, interrupted at intervals by large circular apertures 
through which the object sighted upon is more readily found, as 
shown in the figure. 

111. The spirit-level is a small glass tube, slightly curved 
toward the middle, nearly filled with alcohol, leaving a bubble of 
air in the tube, and closed at both ends. When the level is m a 
truly horizontal position, the bubble of air rests in the middle of 
the tube ; when the level is not horizontal, the bubble seeks the 
more elevated end. When the two levels have each the bubble 
at the middle, each is truly horizontal and their plane is hori- 
zontal ; therefore, the plane of the graduated circle and the 
magnetic needle, which is parallel to the plane of the levels, is, 
also, horizontal. 

112. The brass-head, by which the compass is attached to 
the staff, is furnished with a ball-and-socket joint to give a 
universal motion for purpose of leveling. Sometimes a tripod 
is used as a support instead of a staff, in which case a plumb-bob 
is attached immediately under the centre of the graduated circle 
for the purpose of accurately placing the instrument over any 



76 



ELEMENTS OF SURVEYING. 



[book IIL 



desired point. When a tripod is used, a tripod-head with leyel- 
ing screws (to be described under the transit), instead of a ball- 
and-socket joint, is often used for leveUng. In the description 
which follows, a ball-and-socket joint will be assumed. 




Fio. 59. 



113. To find the bearing of any 
course by the compass. Let PQhe the 
course whose bearing is desired. Place 
the compass exactly over the point P, 
by inserting the staff in the ground at 
that point, or by the plumb-bob if a 
tripod is used. It is usual for a sur- 
veyor to keep the south end of the 
compass towards him and to read the 
bearings from the north end, and this 
position of the compass will be assumed 
in description, unless otherwise stated. 
Turn the sights toward the staff or object at Q. Bring the 
bubbles to the middle of the spirit-levels by the pressure of the 
hand on different parts of the plate. Look through the smaU slit 
in the sight next the person, and turn the compass till the small 
slit in the sight opposite bisects the object at Q, being careful to 
keep the bubbles at the middle of the levels. Let the needle 
come to rest and take the reading indicated by the north end of 
the needle — it will be the angle NPQ, the bearing. When the 
needle is at rest, it lies in the magnetic meridian. The line of 
sights lies in the direction of the course ; when this line lies east 
of the meridian, the bearing is east ; when it lies west of the 
meridian, the bearing is west. When the bearing is east, the 
needle lies to the west of the line of sights, and the reverse ; 
hence, to facilitate the reading the E and W letters on the face 
of the compass are reversed from their natural position. 

In order to prevent a merely mechanical reading, the cover of 



SEC. II.] 



surveyor's compass. 



77 



the compass-box should be unscrewed and a circle of paper 
should be fitted into the bottom of the box so as to conceal the 
letters ; the student will then learn to read from the needle 
alone, the north end of which bears a distinguishing mark or 
color. It is only by such practice that blundering readings can 
be avoided. 



114. To find by the compass the angle subtended at any 
point by two objectp, take the bearing of the course to each 
object, as explained in the last article. The question then is to 
find the angle between any two courses, 
when their bearings are known, which 
may be done as follows : 

Let NS be a meridian passing 
through A. 

Let AB, AC, AH, AD, and AF, 
be five courses running from A, We 
readily deduce the following 




PRINCIPLES, 



^(7 is N 26° W 
AH'\^ ^ 65° W 



When the meridional letters 
are alike, and those of departure 
> also alike, fhe difference of the 
hearings is the angle lettveen the 
courses. 



AB is N 46° E 
^C7is N26° W 
CAB = 72° 



When the meridional letters 
are alike, and those of departure 
unlike, the sum of the tearing s is 
the angle between the courses. 



78 ELEMEKTS OF SURVEYING. [BOOK UL 

When the meridional letters 

^C is N 26° W ') are unlike, and those of departure 

AD is S 66° W V alike, the angle letween the courses 

CAD = 180° -92° = 88° J is equal to 180^, minus the sum of 

the bearings. 



When the meridional letters 
are unlike, and those of departure 
also unlike, the angle hettveen the 
CAF= 180°— 40° = 140° J courses is equal to 180°, minus the 

difference of the bearings. 



^CisN26° W 
AF is S 66° E 



Note. — The above principles are deduced, under the sup- 
position that the two courses are both run from the same 
angular point. Hence, if it be required to apply these rules to 
two courses run in the ordinary way, as we go around the field, 
the bearmg of one of them must be reversed before the calcu- 
lation for the angle is made. 

EXAMPLES. 

1. The bearings of two courses, from the same point, are 
IS" 37° E, and S 85° W; what is the angle included between 
them ? Ans. 132°. 

2. The bearings of two adjacent courses, in going round a 
piece of land, are ]^ 39° W, and S 48° W; what is the angle 
included between them ? A?is. 87°. 

3. The bearings of two adjacent courses, in going round a 
piece of land, are S 85° W, and N 69° W ; what is the angle 
included between them? Ans. 154°. 

4. The bearings of two adjacent courses, in going round a 
piece of land, are :N' 55° 30' E, and S 69° 20' E ; what is the 
anoxic included between them ? Ans, 124° 50'. 



SEC. III.] 



WOBK ON THE FIELD. 



79 



SECTION III. 



WORK ON THE FIELD. 



115. When a piece of ground is to be surveyed, we begin at 
some prominent corner of the field and go entirely around the 
land, measuring the lengths of the bounding lines with the 
chain, and taking their bearings with the compass. It is not 
material whether the ground be kept on the right hand or on 
the left, and all the rules deduced for one of the cases, are 
equally applicable to the other. To preserve uniformity, how- 
ever, in the language of the rules, we shall suppose the land to be 
always kept on the right hand of the 
surveyor. 

Let A BCD be a piece of ground 
to be surveyed, A the point where 
the work is to be begun, and NS a 
meridian. 

On a sheet of paper, rule a single 
column, ^ inch wide, down the middle 
of the left hand page of the note looh, 
as in the example. 




I 



D 
C 
B 
A 



A 

10.00 

A 

7.60 
A 

9.20 
A 
10.40 
A 



(N. 45 E.) 



S. 45i W. 
(N. 35} W.) 

S. 36 E. 
(S. 62 W.) 

K 62 E. 
(S. 31iE.) 



N. 31i W. 



80 ELEME>?^TS OF SURYEYII^G. [BOOK IIL 

Place the compass at A, and take the bearing to B, 
which is FAB ; suppose this angle has been found to be 31^°. 
The bearing from A to B is then N. 31J° W. Enter this bearing 
in the field notes at the right of station A. Then measure the 
distance from A to B, which we will suppose to be 10 ch. 40 1., 
and insert that distance in the column, above the station mark. 

We next take the bearing from B to C, 'N. 62° E., and then 
measure the distance BC = 9 ch. 20 1., both of which we insert 
in the notes as abore. 

At station C we take the bearing to D, S. 36° E., and then 
measure the distance CD = 7 ch. 60 1., and place them in the 
notes. 

At D we take the bearing to A, S. 45J° W., and measure 
the distance DA = 10 ch. We shall then have made all the 
measurements on the field which are necessary to determine the 
contents of the ground. 

116. The reverse-bearing or back-sight, from B to A, is the 

angle ABH \ and since the meridians NS and EG are parallel, 
this angle is equal to the bearing NAB. The reverse-bearing 
is, therefore, S. 31 J° E, and should he entered in the notes in 
parenthesis opposite station B, as in the example. 

The reverse-bearing from C, is S. 62° W.; that is, it is the 
angle ICB = GBC. And generally, 

A reverse-hearing, or hack-sight, should always equal the for- 
ward-hearing, and differ from it only in hoth of the letters hy 
which it is designated. 

117. In taking the bearings with the compass, there are two 
sources of error. 1st. The inaccuracy of the observations ; 2d. 
Local attractions, or the derangement which the needle expe- 
riences when brought into the vicinity of iron-ore beds, or any 
ferruginous substances. 



SEC. III.J 



WORK 01^ THE FIELD. 



81 



To guard against these sources of error, the reverse-bearing 
should be taken at every station ; if this and the forward- 
bearing are of the same value, the work is probably right ; but 
if they differ considerably, they should both be taken again. 

Electricity is a fruitful cause of annoyance in compass work. 
In cold, dry weather, any friction upon the glass cover of the 
needle- box, even that of a cold, dry wind, charges it with elec- 
tricity and the needle no longer traverses freely. To dissipate 
the charge touch the plate with a moistened finger, or breathe 
strongly upon it. If the surveyor uses a pocket lens to read the 
needle, he must see to it that no iron or steel screws are used 
in the casing; nickel-plated mountings are not admissible; 
hard rubber mountings are very troublesome, as they often 
become highly charged, and will drag the needle through 90°, or 
even 180°. Brass, or German silver, are the most satisfactory 
mountings. 

118. In passing over the course 
AB, the northing is found to be HB, 
and the departure, which is west, is 
represented by AJff. Of the course 
BC, the northing is expressed by BG, 
and the departure, which is east, by ivrj 
GC. Of the course CD, the southing 
is expressed by CI, and the departure, 
which is east, by CF. Of the course 
DA, the southing is expressed by KA, 
and the departure, which is west, by 

DK. It is seen from the figure, that the sum of the northings is 
equal to HB + BG = EG ; and that the sum of the southings is 
equal to CI+EA = PA = EG ; hence, the sum of the northings 
is equal to the sum of the southings. 

If we consider the departures, it is apparent that the sum 



Gr- 



Ji 



11 




82 ELEMENTS OF SUEVEYIXG. [BOOK III 

of the eastings is equal to GC-\-CF =^ GF; and that the sum 
of the westings is equal to AH-\-DK ^ GF\ hence, the sum 
of the eastings is equal to the sum of the ivestings. We there- 
fore see, that when the survey is correct, the sum of the northings 
will he equal to the sum of the southvngs, and the sum of the 
eastings to the sum of the westings. 

It would, indeed, appear plain, even without a rigorous 
demonstration, that after having gone entirely round a piece 
of land, the distance passed over in the direction due north 
must be equal to that passed over in the direction due south ; 
and that the distance passed over in the direction due east 
must be equal to that passed over in the direction due west. 

119. The boundaries of a field are generally occupied by 
fences, and frequently also by a border of shrubbery, so that 
chaining along the true boundary is impossible. 

In such cases, it becomes necessary to measure ^, ^^^^ 

an offset at each end of the course (and at right 
angles to it), and of suflBcient length to clear the 
obstructions ; the measurement is then made 
between these temporary stations. 

It is evident that the bearing and length of 
mn, the offset-course, are the same as those 
of MN. 

When such offset-courses are necessary for 
several successive courses, errors are likely to be 
committed, unless the surveyor is careful to make 
new offsets for each course. p^^ ^ 

To survey houndaries LM, MN, NO, &c., along which the 
chaining cannot he done, as in Fig. 64, proceed thus : 

Set the instrument at some point, A for example, far enough 
from LM to clear all obstacles, usually five or six feet; next 



SEC. III.] WOEK Oiq^ THE FIELD. 83 



place a pole at By make Bl)=^Aa by measurement, laying off the 
perpendiculars by the eye alone ; now sight B and note the reading. 
While the compass is sighted on B, look across two notches cut in 
the rim of the compass-box in a line perpendicular to the line of the 
sights and passing through the needle pivot, and fix a plumb-line 
upon LM, as at « ; if the distance Aa, when re-measured, should 
differ much from that used in setting off Bi, the offsets must be 
corrected ; now measure La (a few links only), and, transferring 
the link held at a to A, continue the measurement along AB to 
B ; having set up the compass at B, sight to A for the back- 
sight, or reverse-bearing, and at the same time fix a plumb-line 
at & by "cross-sighting" as before ; transfer the link held at B 
to J), and continue to M, and enter the length of LM, thus 
obtained, in the notes. 

Without moving the instrument, measure the perpendicular 
Be, set a staff at D and make Dd, the perpendicular to MN pro- 
longed, equal to Be ; sight to D for the bearing of JfiV", cross- 
sight to fix c, and do the same at D to fix d', measure Mc, BD, 
dN; Mc-^BD—dN =z MJSf, which enter in the notes. In like 
manner, iVO = iVe + Z>il4-^0. 

120. It has been customary, since the first settlement of this 
country, to use the compass in all land surveys, so that the 
description of lands, in purchase and sale, and by which they 
are recognized in the courts, involves the length and bearing 
of each straight line of the boundary. The method, therefore, 
is, at present, a necessary one. 

The errors to which the compass is liable are so numerous 
and so variable, even in the same instrument, that a change 



84 ELEMENTS OF SURVEYING. [BOOK III. 

of practice is very desirable. Many surveyors, to insure a higher 
degree of accuracy, measure the angles of a field with the 
transit, and then, having determined the bearing of one side 
with sufficient accuracy, calculate the others by a method to be 
shown in a subsequent article. 

121. In surveys of large areas, the surveying party should 
consist of at least four persons — viz., a compass-man, a flag- 
man, and two chain-men. In smaller areas, the work is gen- 
erally performed by the surveyor and one assistant ; the surveyor 
serving alternately as compass-man and hind-chainman, and 
the assistant as flag-man and fore-chainman. 

122. In recording the notes of the survey, the advantage of 
beginning at the bottom of the page is this : that when standing 
on the line to be surveyed, and looking in the direction we 
propose to go, the column in the book lies before us just as the 
line does, and all measurements made to the right or left of the 
line are recorded at the right or left of the column. In surveys 
where many auxiliary notes are taken, a diagram is an important 
aid to a ready interpretation of the other notes. 

GENERAL EXAMPLE. 

. 123. To explain the method, in full, of making a compass 
survey and recording the notes, we will take an example of a 
farm, in which, in addition to the usual survey of the boundary, 
such other measurements are made as to enable us to make a 
correct map of the whole. 

Page 87 represents a farm to be surveyed, and page 86, the 
notes which are made, in the operations on the field. 

Beginning with the corner marked A, the bearing of the 
line AB is taken. In most cases, offsets from both A and B 
would be taken, in order that the survey may be clear of the 
fence, but such offsets are not recorded ; the surveyor must keep 



SEC. III.] WORK ON" THE FIELD. 85 

in mind that it is the boundary of the field that is surveyed, and 
any device by which this is accomplished is no part of his 
notes. 

The record of the bearing of the first course is entered at the 
right of the column (page 86), while the letter designating the 
station, is placed to the left. 

The symbol A, which signifies station, is placed in the 
column, between the letter and the bearing, for each angle of 
the farm. 

In chaining the first course, the intersection of the line 
with any objects worthy of notice is recorded. The first record 
is of the road leading to the quarry. As it is an unimportant 
road, a single measurement of the distance on the course to its 
centre is sufficient to locate it. The distance is 4.30 chains. 

At 11.30 and 12.35 the sides of the turnpike are intersected. 
The bearing of the road, at this point, is also carefully taken and 
recorded. 

The intersections of the garden fence and of the brook are 
also noted (17.40) and (18.10) ; and these, with the entire length 
of the course (31.95), close the record of this line. 

At B, the back-sight upon A is first taken, and entered in 
the notes opposite the ^-station mark, as directed in Art. 116. 
The entry is omitted in the example to avoid croivding and 
confusing the notes. Then the bearing of BC is, taken. Next 
the bearing of the northernmost chimney of the farm-house 
is taken {N. 73° E.). Such bearings serve two purposes. 
They aid in the location of the objects observed, upon the map, 
and serve also, in case of errors, to aid in detecting their 
location. 

In general, in surveying large or small areas, some prominent 
point or points, within the boundary, should be selected, and 
their bearings, from different angles, carefully noted. 



K. 



I 



F 




JE 



D 




£ 




StaJttanA. 



WS7 



A 
/7.S9 
f/.3Q 

^\ 

6.45 
S.40 

A 
7.65 

A 
Id. SI 

A-\- 
^.9 J 
J 3. TO 
/2.90 



8.80 

A'"" 
8.S9 

A 
22.89 

i9.50 

J5.00 
/3.00 

40.00 

7.70 
6.00 

A 
J. 40 

A., 
34.95' 
/3.70 
/8.W 

47.40 

/^.3S 

//.30 
4.30 

A 



fo Stcdi/nvA. 



S30°W 
\FeTiceJT73''W 




SZ7°E 



S7/''£ 
SM^SO'E 



'■••-t%^ 



Fenc&S6d^4JE .^-'iS 



,--' OAKTREE 

W£ASrS/0£ 
OFWHEATFIEUi 



2.00 




ZOO 



7^30°35^ 




S87'>5'W 
Fenc&W^°E 



WS^E 







GarCUrvFeru^ 

TURNPIKE J^ySZS 



ir68''j5>r 

Fig. 65. 



-<^^ 



FARMHOUSE 



MAP OF FARM. 




Fig. 66. 



88 ELEMEi^TS OF SURVEYIN"G. [BOOK III. 

The chimney of the farm-house and the oak-tree in the 
corner of the wheat-field, are thus employed in this survey. 

At C, the corner of the field, is in the centre of the brook, 
and from this point to D, the brook is the boundary. A straight 
line is run between the stations, and offsets are measured to 
each bend of the brook. 

It is necessary, in such a case, for the chainmen to exercise 
unusual care in keeping in the line between the stations, other- 
wise the lengths of the offsets cannot be correctly measured. 

At E, the bearing of the oak-tree is taken (IST. 73|-° E.). On 
the course between E and F, a marsh is encountered, which the 
chainmen pass, by an offset course. 

At F, another bearing is taken of the oak-tree (S. 44J° E.). 

At G, the bearing of the farm-house chimney is noted 
(S. 26° E.). At G and H the bearings of the division-fences are 
taken. On the course from H to /, the turnpike is again 
crossed ; the intersection of both sides, together with the bearing, 
are carefully noted. 

From / to K, the intersection and bearing of the fence 
between the potato and the wheat field, are recorded. The course 
from ^ to ^ closes the survey. 

To locate the buildings about the farm-house, a few measure- 
ments would be necessary ; but tbey may begin with the point 
already located by the bearings taken to the chimney nearest the 
north end of the house. 

The dimensions of the buildings, their distances apart, and 
the direction of one side of each afford sufficient data for locating 
them, correctly, upon the map. 

Note. — The advantage of the compass over other instruments 
with which angles are measured, lies chiefly in this : that the 
Bearing of a course may be measured at any point on the 
line. 



SEC. III.] 



WORK Ol^f THE FIELD. 



89 



When the angle between adjacent sides is taken with the 
Transit, the work can only be done at the corners of the field ; 
and when, as freqnently happens, a hill intervenes between two 
consecutive stations, it becomes necessary to locate a point on 
the hill, in the true line, and then return to the corner to 
measure the angle ; whereas, when the compass is employed, the 
establishment of the intermediate point on the hill affords the 
means of taking the proper bearing without going to the angle. 
Furthermore, the bearings may be measured with the compass, 
by placing it at the alternate stations only. 

The disadvantage of these rapid methods is that there is no 
check upon the needle readings, as there is in back-sighting. 

124. To Correct Local Attraction.— Suppose that bearings 
and reverse-bearings have agreed for several stations, and that 
then a back-sight differs from the preceding fore-sight ; it may 
be concluded that local attraction affects the needle at the last 
station only. 

Suppose the fore-sight at P to have 
been N. 30 E., and the back-sight at S 
to be S. 28 W. ; suppose the fore-sight 
at S to be N. 28 W. ; what is the cor- 
rect fore-sight at iS ? From Fig. 67 
we see that the needle at S, instead of 
being parallel to the needle at P (as 
indicated by the dotted needle) as it 
should be, has been deflected by local 
attraction (as shown by the full needle), 
so that its south end is moved 2° to the W., and its north end 2° 
to the E. of its proper position. Hence the fore-sight at S, being 
taken from the needle in a false position, is 2° too large, and 
should be only K 24 W. By similar reasoning, aided by a 
mental picture of the case, any bearing may be corrected. 




Fig. 67. 



90 



ELEMENTS OF SURVEYING. 



[book III. 



If 110 buck-sight and its fore-sight agree, then take the mean 
of the back-sight and fore-sight which differ least ; or seek some 
ground at a distance, free from local attraction, and then work 
into the plot to be surveyed, correcting each fore-sight on 
the way. 



SECTION IV. 
AREA OR CONTENTS OF GROUND. 

Having explained the necessary operations on the field, we 
shall now proceed to show the manner of computing the contents 
of ground. 

125. The Traverse Table and its Uses. — This table 
shows the latitude and departure coiTesponding to bearings that 
are expressed in degrees and quarters of a degree, from to 90°, 
and for every course from 1 to 100, computed to two places of 
decimals. 

The following is the method of deducing the formulas for 
computing a traverse table ; by means of 
these formulas and a table of natural 
sines, the latitude and departure of a 
course may be computed to any desirable 
degree of accuracy. 

Let AD be any course, and NAD its 
bearing; then AE is the latitude, and ED 
the departure of the course AD, to the bear- 
ing NAD. From Art. 36, we have (using 
nat. sines) 

AD : AE w 1 : cos NAD, 




Fia. 68. 



Whence, 



lat. = course x cosine of bearing. 



SEC. lY.J AREA OR CONTENTS OF GROUND. 91 

And from Art. 32, we have (since sine AED = 1), 
AD \ ED w I \ sine NAD. 

Whence, dep. = course x sine of bearing. 

We have then the following practical rule for computing 
the latitude and departure of any conrse. 

Look in a table of natural sines for the cosine and sine of the 
hearing. Multiply each hy the length of the course, and the first 
product will he the latitude, and the second will he the departure 
of the given course. 

EX A M^LES. 

1. The bearing is 65° 39', the course 69.41 chains: what is 
the latitude, and what the departure ? 

Natural cosine of 65° 39' ...... .41231 

Length of the course ....... 69.41 

Product, which is the Dif. of Lat. . . 28.618437 1 

Natural sine of 65° 39' 91104 

Length of the course . 69.41 

Product, which is the Departure . . . 63.2352864 

2. The bearing is 75° 47', the course 89.75 chains ; what is 
the latitude, and what the departure? 

Natural cosine of 75° 47' 24559 

Length of course 89.75 

Product, which is the Dif. of Lat. . . 22.0417025 

Natural sine of 75° 47 96937 

Length of course 89.75 

Product, which is the Departure . . . 87.0009575 

In this manner, the traverse table given at the end of the 
book, has been computed. When the bearing is given in degrees 



92 ELEMEN^TS OF SUEVEYIN'G. [BOOK III. 

and quarters of a degree, and the difference of latitude and 
departure are required to only two places of decimals, they may 
be taken directly from the trayerse table. 

When the bearing is less than 45°, the angle will be found 
at the top of the page ; when greater, at the bottom. When the 
distance is less than 50, it will be found in the column 
*' distance," on the left-hand page ; when greater than 50, in 
the corresponding column of the right-hand page. 



\T_ 



H .^ 



F 



E 



126. The latitudes or departures of -^ 

courses of different lengths, but which have c 

the same bearing, are proportional to the G 

lengths of the courses. Thus, in the figure, ^^ 
the latitudes AG, AC, or the departures 
GF, CB, are to each other as the courses ^ 

AF, AB. Fig. 69. 

Therefore, when the distance is greater than 100, it may be 
divided by any number which will give an exact quotient less 
than 100 ; then the latitude and departure of the quotient being 
found and multiplied by the divisor, the products will be the 
latitude and departure of the whole course. It is also plain, 
that the latitude or departure of two or more courses, having 
the same bearing, is equal to the sum of the latitudes or depar- 
tures of the courses taken separately. 

It is always better to obtain the lat. and dep. by addition 
than by multiplication ; thus, if we sought the lat. and dep. of a 
course of 190 ft., bearing 36°, they would be found by multiplica- 
tion thus, 

lat. 19 X 10 = 15.37 x 10 = 153.70 
dep. 19 X 10 = 11.17 X 10 = 111.70 

giving results containing ten times the error of lat. and dep. of 
19, as given in the table. By addition we should have 



I 



SEC. IV.] AREA OR CONTENTS OF GROUND. 93 

lat. lOO + lat. 90 = 80.90 + 72.81 = 153.71 
dep. 1^0 + dep. 90 = 58.78 + 52.90 = 111.68 

which are closer approximations than the former. 

Hence, we should always make our multipliers as small as 
possible. 

EXAM PLE S. 

1. To find the latitude and departure of 614, to bearing 29|-°. 



Latitude of 100 x 6 = 522.24 
Latitude of _14 = 12.18 

Latitude of 614 534.42 



Departure of 100x6 = 295.44 
Departure of 14 = 6.89 

Departure of 614 = 302.33 



2. To find the latitude and departure for the bearing 62J°, 



aud the course 7855 chains. 

Latitude for 7800 . 3602.00 
Latitude for 55 . 25.40 

Latitude for 7855 . 3627.40 



Departure for 7800 . 6919.00 
Departure for 55 . 48.79 

Departure for 7855 . 6967.79 



Note. — When the distances are expressed in whole numbers 
and decimals, the manner of finding the latitudes and departures 
is still the same, except in pointing off the places for decimals ; 
but this is not difficult, when it is remembered that the column 
of distances in the table may be regarded as decimals, by 
simply removing the decimal point to the left in the other 
columns. 

3. To find the latitude and departure for the bearing 47f°, 
and the course 37.57. 



Latitude for 37.00 
Latitude for .57 

Latitude for 37.57 



24.88 
.38 

25.26 



Departure for 37.00 
Departure for .57 

Departure for 37.57 



27.39 
.42 

27.81 



94 



ELEMENTS OF SURVEYIXG. 



[book III. 



127. Balancing the "Work. — The field-notes having been 
completed, rule a new table, as below. 

Then find, from the traverse table, the latitude and departure 
of each course, and enter them in the proper columns opposite 
the station. 

Then add the column of northings, and also the column 
of southings; the two sums should be equal to each other. 
If they are not, subtract the less from the greater ; the remainder 
is called the error in latitude. Find the error in departure in 
the same way. 

This error for latitude or departure must be distributed 
among the latitudes or departures of all the courses, in propor- 
tion to the length of each course, observing to add the correction, 
when applied to the deficient column, and to subtract it, when 
applied to the other. 

This may be illustrated by the example of (Art. 115). 



stations. 


Bearings. 


Dis. 


Dif. Lat. 


Dep. 


Balance. 


:n". 


S. 


E. 

+ 


W. 


Lat. 


Dep. 


A 


N. 311° w. 


10.40 


8.87 


.... 


.... 


5.43 


+ 8.86 


-5.44 


B 


K62° E. 


9.20 


4.32 




8.13 




+ 4.31 


+ 8.12 


C 


S. 36° E. 


7.60 




6.15 


4.47 




—6.15 


+ 4.46 


D 


S. 451° w. 


10. 




7.01 




7.13 


-7.02 


-7.14 


Sum of courses, 37.20 


13.19 


13.16 


12.60 


12.56 








13.16 




12.56 









Error in latitude 



.03 



.04 Error in departure. 



The error in latitude, 3 links, is to be distributed among the 
northings and southings, in proportion to the lengths of the 
courses ; a part to be added to the southings, and the remaining 
part subtracted from the northings. The error in departure is 
similarly distributed among the eastings and westings. For this, 



SEC. IV.] AREA OR CONTEi^^TS OF GROUND. 95 

two new columns are formed, called, the balanced latitudes and 
departures ; and to these columns the latitudes and departures 
are transferred, after the corrections have been made; the north 
latitudes being marked -f , and the south latitudes — , in order 
to distinguish them readily, and also, for convenience in the 
calculations which follow. 

The error of .03 in the latitudes is distributed among the 
latitudes, by subtracting 1 link from each of the northings of 
courses A and B, and adding 1 link to the southing of course D. 
This produces a balance. 

Of the error of 4 links in the departures, 1 link is added to 
each of the departures west, and 1 link subtracted from each 
of the departures east. This produces a balance. 

Note. — When a knowledge of the conditions under which 
the survey was made, enables us to determine that errors were 
more likely to occur at certain points, it is best to apply the 
corrections to those courses where it seems probable the errors 
were made. 

128. The limit of error to be allowed depends, of course, 
upon the importance of the survey. 

In ordinary farming districts, the error should be as small 
as 1 link to 5 or 10 chains of perimeter. 

The "error of the survey" should be considered as the length 
of the line necessary to dose tlie boundary, and is equal to the 
square root of the sum of the squares of the errors of latitude 
and departure. Thus, in the above example, the error of the 
survey is 5 links. The perimeter being 37.20 chains, the error 
is about 1 link to 7.45 chains, or -^ of the perimeter. 

129. It will be well to bear in mind the fact, that if the 
error in the perimeter has been made in one course only, and 
distributed, by the ordinary methods of balancing, among all the 



96 ELEMENTS OF SURVEYING. [BOOK IIL 

courses, the error in area will be larger than the error in 
perimeter. 

130. When the error is so large that a re-survey becomes 
necessary, the halancing should be carefully re-examined. In 
many cases, the location of the error may be determined 
by inspection of the computation, and a portion of the labor 
of a re-survey thereby saved. 

This refers more particularly to those cases where the error 
is one of chaining, and is mostly in one course. Errors of this 
kind occur sometimes with experienced chainmen, who draw 
the chain properly between the courses, but make occasionally 
an error in counting the fractional part of a chain at the end 
of a course. In such cases, the location of the error may be 
detected by observing, first what columns contain errors, and 
secondly the ratio of the errors of Latitude and Depari:ure. 

When the eiTor in the survey has been a single one, of dis- 
tance onlv, then the ratio between the errors of Latitude and 
Departure must be the same as the ratio between the Latitude 
and Departure of the course to be corrected. If the errors be 
in northings and westings, then the courses running either 
North and West, or South and East, should be examined. 

131. The surveyor should take every possible precaution 
igainst en'ors in the bearings. This is accomplished by back- 
sighting, taking bearings of some one object from several sta- 
tions, and also by taking bearings of stations across the field. 
These precautions wiU give, in general,, sufficient data for the 
detection of an error in hearing ; for, by mapping the survey 
and drawing the lines to indicate the extra bearings, the error 
is revealed by the failure of the Hnes to meet at a common 
point. 






SEC. IV.] AREA OE CONTENTS OF GROUND. 97 

132. One source of error, m large surveys with the compass, 
is frequently overlooked. This is the diurnal variation ; there 
is sometimes as much as 15 minutes variation during the day- 
light hours. 

Errors from this source can only be avoided by testing the 
compass, at intervals of two or three hours, by taking the 
bearing of the same line. 

133. There is one kind of error frequently made in reading 
the compass when the bearing is nearly east or west. The 
error arises from reading North for South, or the reverse. If 
the survey is otherwise correct, the error in latitude is just 
twice the latitude of the course containing the error. 

DOUBLE MERIDIAN DISTANCES 

134. After the work has been balanced, the next thing to 
be done is to calculate the double meridian distance of each 
course. 

For this purpose, any meridian line may be assumed. It is, 
however, most convenient to assume that meridian which passes 
through the most easterly or westerly station of the survey ; 
and these two stations are readily determined by inspecting 
the field-notes. 

Having chosen the meridian, let the station through which 
it passes be called the principal station, and the course which 
begins at this point, the first course. Care, however, must he 
taken not to confound this luith the cotcrse which begins at 
station 1, and which is the first course that is entered in the 
field-notes. 

It has already been remarked (Art. 105), that all departures 
in the direction east are considered as plus, and all departures 
in the direction west as minus. 



It 



98 



ELEMENTS OF SURVEYING. 



[book III 




M C 



Fio. 70. 



135. To deduce a rule for finding the double meridian dis- 
tance of any course. Let SN be the assumed meridian. Let 
BG represent any course, and AB the preceding course ; also, 
let D and B be their middle points. Draw BH, BG, and CM, 
perpendicular to the assumed meridian 
NS. Draw also Al, EK, and BB, par- 
allel to N8. Then '2DG is the double 
meridian distance of the course BC, and 
"^EH = 2KG, is the double meridian 
distance of the course AB. 

Now, 2DG = 2GK+ 2KB + 2BB; 
but 2KB = IB is the departure of the 
course AB, and 2BD = MC is the de- 
parture of the course BC ; consequently, 

2GD = 2GK + IB + MC\ 

hence, the double meridian distance of a course is equal to 
the double meridian distance of the preceding course, plus the 
departure of that course, plus the departure of the course itself : 
if there is no preceding course, the first two terms become 
zero. We therefore have the following 

EuLE. — L The double inej^idian distance of the first 
course is equal to its depaHure ; 

II. Tlve double meridian distance of the second course 
is equal to the double meridian distance of the first course, 
■plus its departure, plus the departure of the second course ; 

III. Tixe double meridian distance of any course is 
equal to the double meridian distarice of the preceding 
course ; plus its departure, plus the departure of the 
course itself. 

Note.— It should be recollected that plus is here used in 
its algebraic sense, and that, when the double meridian distance 



SEC. IV.] 



AREA OR CONTEi^^TS OF GROUKD. 



99 



of a course, and the departure which is to be added to it, 
are of different names, that is, one east and the other west, 
they will liave contrary algebraic signs ; hence, their algebraic 
sum will be expressed by their numerical difference, with the 
sign of the greater prefixed. 

If the assumed meridian cuts the enclosure, the double 
meridian distances estimated to the east are plus, and those 
on the west must be taken with the minus sign. 

The double meridian distance of the last course should be 
equal to the departure of that course. A verification of the 
work is therefore obtained, by comparing this double meridian 
distance with that departure. 

AREA. 



136. Let us resume the example 
of Art. 115. We will first write the 
differences of latitude and the double 
meridian distances of the courses, 
in the following table: 




Fig. 71. 



Stations. 


Dif. of Latitude. 


D. M. D. 


Area. 

+ 


Area. 


A 


4- cB 


+ %l)a 


^cAB 




B 


+ Bs 


+ ^p 


2BsC 




C 


-Dy 


-h 2>iA 




2ms CD 


D 


-Df 


+ 2ed 




2cmDA 



It is evident, that cB multiplied by 2ba, or cA, will give 
double the area of the triangle cAB. But cB and ba are both 



L.ofC. 



100 ELEMENTS OF SURVEYIITG. [BOOK III. 

plus ; hence, the product will be plus, and must be put in 
the column of plus areas. Double the area of the triangle 
BsG, is equal to Bs multiplied by Hqp, wliich product is also 
plus. 

The area of the trapezoid msCD is equal to yD, or ms, multi- 
plied by nh (Geom., Bk. IV, Prop. VII) ; hence, double the 
area is equal to yD into 2nh, But since yD (being a southing) 
is minus, and 2nh plus, it follows that the product will be 
negative; hence, it must be placed in the column of negative 
areas. 

Double the area of the trapezoid cADm, is equal to Df, or mc, 
multiplied by 2de ; but, since Df is negative and 2de positive, 
the product will be negative. 

It is now evident that the difference between the two columns 
is equal to twice the contents of the figure ABOD ; and since 
the same may be shown for any other figure, we have, for finding 
the areas, the following general 

EuLE. — I. Multiply the double meridian distance of 
each course by its northing or southing, observing that like 
signs in the multiplicand and multiplier give plus in the 
product, and that unliJce signs give minus in the product. 

II. Place all the products luhich have a plus sign in 
one column, and all the products which have a minus 
sign in another. 

III. Add the columns separately and take the dif- 
ference of their sums ; this difference will be double the 
area of the laitd. 

Note. — When offsets are measured, the figures of which 
the offsets are bounding lines are, practically, triangles and 
trapezoids, and the areas of these are to be separately computed, 
and added to or subtracted from, as the case may be, the area 
obtained by the foregoing rule. 



SEC. IV.] 



AEEA OR CONTENTS OF GROUND. 



101 



137. We will now make the calculations of this example, in 
numbers, from the field-notes, which are the following : 



stations. 


Bearings. 


Distances. 


Diff. Lat. 


Dep. 


D. M. D. 


A 


N 31J° W 


10.40 


+ 8.86 


—5.44 


+ 18.02 
— 7.14 
-5.44 


+ 5.44 


*B 


N62° E 


9.20 


+ 4.31 


+ 8.12 


+ 8.12 


C 


S 36° E 


7.60 


-6.15 


+ 4.46 


+ 8.12 
+ 4.46 

+ 20.70 


D 


S 45i° W 


10. 


— 7.02 


— 7.14 


4.46 
-7.14 


+ 18.02 



We see, from inspecting the notes, that B is the most 
westerly, and D the most easterly station. Either of them may, 
therefore, be taken for the principal station. Let us assume B 
for the principal station, through which the assumed meridian 
passes, and distinguish it by a star, thus *. 

Having done so, we enter the departure 8.12 in the column 
of double meridian distances, which is the double meridian 
distance of the course from B to C. The double meridian 
distances of the other courses are calculated according to the 
rule ; and as the last, which is that of the course from A to 
B, is equal to the departure of that course, the work is known 
to be right. 

Let us now form a new table, which will complete the 
arithmetical part of the work. 



102 



ELEMENTS OF SUEVETIXG. 



[book in. 



Sta. 


Bearings. 


Dist. 


Dif. Lat. 


D. M. D. 


Area. 

+ 


Area. 


A 


X 31^° W 


10.40 


+ 8.86 


+ 5.44 


48.1984 




*B 


N62° E 


9.20 


+ 4.31 


+ 8.12 


34.9972 




C 


S 36° E 


9.60 


— 6.15 


+ 20.70 




127.3050 


D 


S 45i° W 


10. 


— 7.02 


+ 18.02 




126.5004 












83.1956 


253.8054 



Area in sq. ch. 
A71S. SA. 2 B. 4.88 P. 



83.1956 

2 ) 170.6098 
85.3049 



PLOTTING. 

138. To make a plot of the 
ground, draw any line, as WS, to 
represent the meridian passing 
through the principal station ; and 
on this line take any point, as B, 
to represent that station. Beginning 
at B, take the algebraic sum of the 
balanced latitudes and, also, of the 
balanced departures of the stations 
following. These sums are, respec- 
tiyely, the total latitudes and total 
departures (or longitudes, see Art. 148, Note) of the several 
courses with respect to the principal 
station, B. Tabulate the results (see 
example, p. 94) as shown in the margin. 
Through B, draw BX (Fig. 72) per- 
pendicular to NS. On XS lay off 
{upward since 4.31 is plus) Bs e;^ual 
to 4.31 units of the scale on which the 




Fig. 72. 



sta. 

*B 
C 
D 
A 


Total lat. 
from B. 


Total dep. 
from B. 


0.00 

4.31 

-1.84 

-8.86 


0.00 

8.12 

12.58 

5.44 



SEC. IV.] AREA OR CONTENTS GROUND. 103 

plot is to be made; and on BX lay off (to the right since 
8.12 is plus) Bf equal to 8.12 units of the scale. From s, as a 
centre, with Bf, or 8.12, as a radius, describe an arc ; and 
from /, as a centre, with Bs, or 4.31, as a radius, describe 
an arc ; the intersection of these arcs will be the position on 
the paper of Station 0. On N8 lay off [downiuard since 1.84 
is minus) Br equal to 1.84 units ; and on BX lay off to the 
right Bq equal to 12.58 units. From r, as a centre, with 
Bq, or 12.58, as a radius, describe an arc ; and from q, as a 
centre, with Br, or — 1.84, as a radius, describe an arc; the 
intersection of these arcs will be the position on the paper of 
Station D. Determine, in like manner, the position on the paper 
of each successive station ; connect the points so determined by 
straight lines, and a complete plot of the ground will be obtained. 
As a check, the distances BC, CD, &c., should be measured on 
the plot. 

139. It is convenient, but not necessary, to take the most 
easterly or most westerly station as the principal station. It 
is simply to be remembered that north latitudes and east 
departures Sireplus, and south latitudes and west departures are 
minus ; that total latitudes and total departures are algebraic 
sums ; that plus latitudes are to be laid off on the meridian, from 
the principal station, upward, and minus latitudes dotumvard ; 
that plus departures are to be laid off on the east and west 
line, from the principal station, to the right, and minus depar- 
tures to the left. 

There are, of course, other metliods of plotting; but the 
one here given is, perhaps, the best m ease and accuracy of 
execution. If the latitudes and departures have been correctly 
balanced, and the drawing carefully made, the survey will 
'certainly "close." 



104 



ELEMENTS OF SURVEYING. 



[book III. 



140. 



-EXAMPLES. 



1. It is required to determine the contents and plot of a piece 
of land, of which the following are the field-notes, viz.: 



Sta- 
tions. 

A 


Bearings. 


Dist. 


Dif. Lat. 


Dep. 


Balanced. 


D.M.D. 

+ 


Area. 

+ 


Area. 


+ 


s 


£ 

+ 


w 


Lat. 


Dep. 


N46rW 20.76 


14.29 






15.06 


+ 14.30 


-15.04 


15.04 


215.0720 




B 


N51f°E 13.80 


8.54 




10.84 




+ 8.55 


+ 10.86 


10.86 


92.8530 




C 


E 21.35 






21.35 






+ 21.37 


43.09 






D 


S56° E 27.60 




15.44 


22.88 




—15.43 


+ 22.90 


87.36 




1347.9648 


E 


S 331° W 18.80 




15.72 




10.31 


—15.71 


—10.29 


99.97 




1570.5287 


F 


N 741° W, 30.95 


8.27 






29.83 


+ 8.29 


-29.80 


59.88 


496.4062 








31.10 


31.16 


55.07 


55.20 


804.3312 2918.4935 




31.10 


55.07 


804.3312 




Error . . .06 


.13 


Error. 2)2114.1623 
















Am. 


105 ^. 2 i?. 33 P. 


1057.08115 



PLOT OF THE GROUND. 




Fig. 7a 

KoTE. — When the bearing is due East or due West, the error 
in latitude is nothing, and the corrections for latitude must be 
distributed among the other courses. So, when the bearing is 
due North or due South, the error in departure is nothing, and 
the error in departure must be distributed among the other 
courses. In the examples for practice, we have not been as 
careful to have as close balances as must be had in actual work 
on the field. 



SEC. IV.] 



AREA OR CONTENTS OF GROUND. 



105 



2. Eequired the contents and plot of a piece of land, of 
which the following are the field-notes. 



Stations. 


Bearings. 


Distances. 


A 


S 34° W 


3.95 ch. 


B 


S 


4.60 


C 


S 36i° E 


8.14 


D 


N 59J° E 


3.72 


E 


]Sr25° E 


6.24 


F 


N16° W 


3.50 


G 


N65° W 


8.20 



Ans. 10 A. E. 5 P. 

3. Required the contents and plot of a piece of land, from 
the following field-notes. 



stations. 


Bearings. 


Distances. 


A 


S 40° W 


70 rods. 


B 


]Sr45° w 


89 


C 


N36°E 


125 


D 


N 


54 


E 


S 81° E 


186 


F 


S 8° W 


137 


G 


W 


130 



Ans. 207 A. 3 E. 33 P. 



106 



ELEMENTS OF SURVEYING. 



[book III. 



4. Required the contents and plot of a piece of land, from 



the following notes. 



Stations. 


Bearings. 


Distances. 


A 


S 40i° E 


31.80 ch. 


B 


N54° E 


2.08 


C 


N 29J° E 


2.21 


D 


N 28i° E 


35.35 


E 


N57° W 


21.10 


F 


S 47° W 


31.30 



Ans, 92 A. 3 R. 32 P. 

5. Required the area of a survey, of which the following 
are the field-notes. 



Stations. 


Bearings. 


Distances. 


A 


N'42°E 


5.00 ch. 


B 


East. 


4.00 


C 


N 9^E 


4 00 


D 


S 69° E 


5.56 


E 


S 36° E 


7.00 


F 


S 42° W 


4.00 


G 


S 75° W 


10.00 


H 


N39° W 


7.50 



If, in this example, we assume A as the principal station, 
the double meridian distances will all be plus, and the posi- 
tive area will exceed the negative. 

In balancing, we shall find the error in southing to be 
.28 ch., and in westing .22 ch. The area is 13 A. B. 11 F, It 
should, however, be remarked, that in all the examples the 
answers may be slightly varied by distributing the corrections. 



SEC. IV.] 



AREA OR CONTENTS OF GROUND. 



107 



6. What is the area of a survey of which the foUowiug 
are the field-notes ? Make the plot. 



Stations. 


Bearings. 


Distances. 


A 


N 75° 00' E 


54.8 rods. 


B 


N 20° 30' E 


41.2 


C 


East. 


64.8 


D 


S 33° 30' W 


141.2 


E 


S 76° 00' W 


64.0 


F 


North. 


36.0 


G 


S 84° 00' W 


46.4 


H 


N 53° 15' W 


46.4 


I 


N 36° 45' E 


76.8 


J 


N 22° 30' E 


56.0 


K 


S 76°45'E 


48.0 


L 


S 15° 00' W 


43.4 


M 


S 16° 45' W 


40.5 



In this survey D is the most easterly and I the most westerly 
station. The area is equal to 110 A. 2 R. 23 P. It may yary a 
little, on account of the way in which the balancing is done. 

7. What is the area of a survey of which the following 
are the notes? Make the plot. 



Stations. 


Bearings. 


Distances. 


A 


S 461° E 


80 rods. 


B 


S 51}° W 


55.20 


C 


West. 


85 


D 


N56° W 


110.40 


E 


N 33i° E 


75.20 


F 


S 74i° E 


123.80 



Ans. 104 A. 1 R. 16 P. 



108 



ELEMENTS OF SURVEYING. 



[book III. 



8. Eequired the area of the farm, of which the survey 
notes are given in Art. 123. 



stations. 


Bearings. 


Distances 
(in chains). 


A 


N" 68° 55' W 


31.95 


B 


N 8° E 


1.40 


C 


S 87° 05' W 


22.89 


D 


N 30° 35' E 


8.39 


E 


N 4^° 05' E 


23.91 


F 


N 87° 05' E 


14.51 


G 


S 64° 30' E 


7.55 


H 


S 71" E 


12.21 


I 


S 27° E 


17.39 


K 


S 30° W 


16.97 



Offsets between C and 
D to be added, 

15.9450 chains. 

Offsets to be subtract- 
ed, 15.1885 chains. 



9. To determine the bearing and distance from one point to 
another, when they are so situated that one cannot be seen 
from the other. 

Let A and C he the two points, and AB a, 
meridian passing through one of them. From 
either of them, as A, measure a course A 2, 
of a convenient length in the direction toward 
C, and take the bearing with the compass. 
At 2, take the bearing of a second course, and 
measure the distance to 3. At 3, take a third 
bearing and measure to 4. At 4, take the bearing to (7, and 
measure the distance from 4 to C. 

Then, the difference between the sum of the northings and 




Fig. 74. 



SEC. IV.] 



AEEA OR CONTENTS OF GROUND. 



109 



the sum of the southings will be denoted by AB ; and the 
difference between the sum of the eastings and the sum of the 
westings, by BC. The base AB, and the perpendicular BC of 
the right-angled triangle ABO, are then known. The angle 
at the base, BAC, is the bearing from ^ to C; or the equal 
alternate angle at C is the bearing from C to A, and the 
hypothenuse AC, is the distance. 

Having measured the bearings and courses on the field, 
form a table, and find the base and perpendicular of the right- 
angled triangle, in numbers ; after which, find the bearing and 
distance. 



stations. 


Bearings. 


Distances. 


N. 


s. 


E. 


w. 


A 


N 61° W 


40 ch. 


19.30 






34.98 


B 


N42° W 


41. 


30.47 






27.43 


C 


^12° E 


16.10 


15.75 




3.35 




D 


N47°E 


32.50 


22.16 




23.77 








AB = 


.87.77 




27.12 


62.41 



27.12 



05 = 35.29 ch 



To find the angle BAC, or the bearing from A to C. 

Kadius : tan A : : AB : BC, 

or, AB : BC :: B : tan ^ ; 

that is, applying logarithms, 

(a. c.) log ^^ (87.77) 8.056654 

log ^(7 (35.29) 1.547652 

log i? 10. 

log tan J 21° 54' 12" 9.604306 



110 ELEMEi^TS OF SURVEYING. [BOOK III. 

To find the distance AC. 

^m A \ R '.: BC : AG) 
Applying logarithms, 

(a. c.) log sin A 21° 54' 12" 0.428242 

logi^ 10. 

log BG (35.29) 1.547652 

log ^C' 94.6 1.975894 

Hence, the bearing and distance are both found. 

Note. — Had any of the courses run south, AB would have 
been equal to the sum of the northings, minus the sum of 
the southings. 

141. The last problem affords an easy method of finding 
the bearing and length of one of the courses of a survey, when 
the bearings and lengths of all the others are known. It may 
be necessary to use this method when there are obstacles which 
prevent the measuring of a course, or when the bearing cannot 
be taken. Indeed, two omissions may in general be supplied by 
calculation. It is far better, if possible, to take all the notes on 
the field, and it is necessary to do so, if accuracy is required. 
Tor, when any of them are supplied by calculation, there are 
no tests by which the accuracy of the work can be ascertained, 
and all the errors of the notes affect also the parts which are 
supplied. 

If necessary, however, the following omissions in the field- 
notes may be supplied by calculation, viz : 



SEC. IV.] 



AREA OR COI^TEN-TS OF GROUN"D. 



Ill 



I. The bearing and the length of one course. 
142. The following are the field-notes of a survey: 



stations. 


Bearings. 


Distances. 


A 


N60° W 


9.72 ch. 


B 


N 17i° E 


7.65 


C 


N 15|° W 


9.40 


D 


N 63|° E 


10.43 


E 


S 49° E 


8.12 


F 


S 13^° E 


8.45 


G 


S 16f°E 


6.44 


H 







With the known bearings and distances, find the correspond- 
ing latitudes and departures, as in annexed table : 



stations. 


Bearings. 


Distances. 


N. 


s. 


E. 


w. 


A 


N60° W 


9.72 


4.86 






8.41 


B 


N 17i° E 


7.65 


7.31 




2.27 




C 


N 15f° W 


9.40 


9.05 






2.55 


D 


N 63f° E 


10.43 


4.61 




9.36 




E 


S 49° E 


8.12 




5.33 


6.13 




F 


S 131° E 


8.45 




8.22 


1.98 




G 


S 16|° E 


6.44 




6.17 


1.86 




H 




















25.83 


19.72 


21.60 


10.96 








19.72 




10.96 





Deficiency 



Denciencv /^ 1 1 
Lat. South. 0«-Li- 



^0 P,4- Deficiency 
XU. U* in Dep. West. 



The latitude of the course HA in the plot (Fig. 75), must. 



12 



ELEMENTS OF SUEYEYING. 



[book IIL 




therefore, be 6.11, and its departure 10.64; hence, in the right- 
angled triangle HP A, 

PA = 6.11, 

HP = 10.64; 

The angle PAH = angle QHA, 

or the bearing of the course HA. 

Find PAH, as in preceding ex- 
ample, thus, 

AP : PH :: R \ tan PAH; 

that is, by logarithms, 

(a.c.) logJP(6.11) = 9.213959 

log P^ (10.64) = 1.026942 

logi^ = 10. 

log tan PAH = 10.240901 

Hence, PAH is 60° 8', and since the deficiencies were in south 
lat. and west dep., the bearing of HA is S. 60° 8' W. 

To find the length of the course, we have, 

sin PAH : R :: PH : HA ; 

that is, by logarithms, 

(a. c.) log sin PAH {60° 8') = 0.061887 
\ogR = 10. 

log PH = 1.026942 

log^^ = 1.088829 

Hence, HA = 12.27 ch., which is the length of the required 
course. HA. 



SEC. lY.] AKEA OK COKTENTS OF GROUND. 

1. In a survey we haye the following notes : 



113 



stations. 


BeariDgs. 


Distances. 


A 


N 31J° W 


10 ch. 


B 


N 62}" E 


9.25 


C 






D 


S 45J° W 


10.40 



What is the bearing and distance from station C to D ? 

( Bearing, S 38" 52' E. 
^^' I Distance, 7.03 ch. 



2. In a survey we have the following notes : 



stations. 


Bearings. 


Distances. 


A 


S 40i° E 


31.80 ch. 


B 


]Sr54° E 


2.08 









D 


N 28f E 


35.35 


E 


N57° W 


21.10 


F 


S 47° W 


31.30 



What is the bearing and distance from C to D ? 

, Bearing, JST 34° 47' E. 
Ans. i 

Distance, 2.19 ch. 



II. The bearing of one course and the length of another, when 
the courses are (1) contiguous; (2) separated. 

143. First. — ^Example when the courses are contiguous. In 
Kg. 75, let the bearing of DE and the length of EF be required, 
the bearings and lengths of the other courses being known. 



114 



ELEMENTS OF SURVEYING. 



[book III. 



As before, find latitudes and departures for known bearings 
and distances, obtaining annexed table : 



stations. 


Bearings. 


Distances. 


N. 


S'. 


E. 


w. 


A 


N60° W 


9.72 


4.86 






8.41 


B 


N17i° E 


7,65 


7.31 




2.27 







N15f° W 


9.40 


9.05 






2.55 


D 




10.43 










E 


S 49° E 












F 


S 131° E 


8.45 




8.22 


1.98 




G 


S 16f° E 


6.44 




6.17 


1.86 




H 


S 60° 8' W 


12.27 




6.11 




10.64 



Draw a line DF and find the bearing and the length of DF 
in the plot ABCDFOH, as in Art. 142. The length will be 
found to be 15.506, and the bearing 87° 20' 19" ; the angle DFE 
is then equal to 87° 20' 19"— 49° (the bearing of EF) = 38° 20' 
19" ; then in the triangle DBF, the sides DF (10.43), and FF 
(15.506), and the angle DFF (38° 20' 19") are knowii, and the 
angle DFF will be found from the proportion, 

DF : DF :: sin DFF : sin DFF 

to be equal to 112° 45' ; hence, the bearing of DF is 112° 45'— 49° 
= 63° 45'. The length of the course ^i^ will be found from the 
proportion, 

sin DFF{^^° 20' 19") : sin FDF (28° 54' 41") : : i)^ (10.43) : FF 

to be equal to 8.12. 

Note. — In finding the angle DFF doubt arises as to the 
proper angle, corresj)onding to the logarithmic sine, to be taken ; 
DFFmsij be either 67° 15', or 112° 45' (see Art. 39); and hence 
the bearing of DF may be either 18° 15, or 63° 45', without 



SEC. IV.j 



AREA OR CO:NtTEKTS OF GROUND. 



115 



affecting the length of the given course DE. If the smaller 
angle 67° 15' is taken as value of DBF, then angle FDE is 74° 
24' 41", instead of 28° 54' 41", and the length of the required side 
EF, as determined from the proportion, 

sin DFE : sin FDE :: DE : EF 

will be 16.19, instead of 8.12, without affecting the given bearing. 
When the length of the side EF, whose bearing is required, is 
not greater than that of the auxiliary line DF, the method given 
is of no service, unless some independent means exist of 
determining which of the two possible solutions is the one to be 
taken. 

The following are the notes of a survey : 



Stations. 


Bearings. 


Distances. 


A 


North 


12.84 ch. 


B 


N32°E 


17.82 


C 


N80°E 


24. 


D 


S 48° E 


27. 


E 


S 18° W 




F 




46.21^ 



Eequired the length of the course EF, and the bearing of the 
course FA. 

Length of EF = 28.60. 

Bearing of FA = N 73° 28' 21" W. 

144. Secon'D. — Example when the courses are separated. In 
Fig. 75 let the bearing of AB and the length of DE be unknown. 

From B, Fig. 76, one extremity of the course whose bearing 
is unknown, draw BP parallel and equal to DE, whose length is 
required. Draw then PQ and QE, equal and parallel to ^Cand 
CD, respectively; and draw PA. The lengths and bearings of 



116 



ELEMENTS OF SURVEYING. 



[lU)OK III. 



the courses PQ luul Ql' arc the Biimo 

as those of BC \\\\(\ ('/>, respectively; 

hence, in the li«,nne A r Q h' I'V H iha ^\ 

lengtlis and hearin<;s of :ill the courses 

except A /' are known ; and the length 

and boariiii; of A r may be found as 

in Art. 142. Then in (he triangle 

APB, the length of A P i\m\ A B are 

known, and (he angle .17V>' (whicii is 

the sum or di (Terence of the bearings 

of BPaml AP) ; hence, as in Art. 143, 

the length of BP or its equal DE, and the bearing of A B may bo 

found, ((lillespie's Land Surveying, j). 21)!)). The same doubt 

as to the proper solution exists here, as in the previous 

example. 



n 




111. The bearings of two courses when they are (1) contiguous; 

(2) separated. 

145. First. — Example when they are contiguous. In T'ig. 
75, let the bearings of DE and EF be unknown. Draw a line 
from D io /' and find its length and bearing from the lati(udes 
and departures of the other courses as belbre. Then in the triangle 
DEF the three sides are known and the three anjilos may be found 
as in Ar(icle41. From the angle ^DjP and (lie b(^aring of DF, 
the bearing of DE may be found; and from llie angle EFD ami 
the bearing of DF, the bearing of EF may bo found. 



146. Second. — Example when the courses are separated. Let 
AB and BE, 1^'ig. 75, be (lie courses whose bearings are un- 
known. Assume the ligure as in Eig. 7(). ^Phon (ho bearing 
and length of A r may be found as before. In the (riangle APB, 
all the sides are known and the angles may be found as in the 



SEC. IV.] 



AREA OR CONTENTS OF GROUND. 



117 



\l 



preceding article. Then the bearing of BP, or its parallel DE, 
may be found from the angle BPA and the bearing of FA ; and 
the bearing of AB, from the angle PAB and the bearing of AP 
(Gillespie's Land Surveying). 

IV. The lengths of two courses. 

147. * Let A = bearing of the first 
unmeasured course, FG, of the survey. 

B = bearing of the second course, 
GIL 

= angle between the courses. 

L = total latitude, and 

Z> =: total departure of the two un- 
measured courses, obtained by comput- 
ing the total latitudes and total departures 
of the measured courses, UK to PF, as in Art. 142. 

Let X = length of the first course. 

y =z length of the second course. 

We shall then have, 

L = (lat. ic+lai y) =^ x cos A-\-y cos B, 
D = (dep. x-{-d(i]). y) = a; sin A-{-y sin B. 

Multiply (2) by cot B, 

D cot B ^^ X sin A cot B-\-y cos B» 

Subtract (3) from (1) and solve with regard to x, 

L—D cot B 




Fig 77. 



(1) 

(3) 



X 



cos ^— sin A cot B 
Multiply both numerator and denominator by sin B, 



• Prot H. 8. Manroe. E.M., Ph.D., and J. Woodbridge Davbs, C.E., Ph.D., In " School of 
Minee Quarterly," for November, 1881. 



Ifc 



118 



ELEMENTS OF SURVEYING. 



[book III. 



X = 



L siu B—D COS B 
sm {A—B) 

or, substituting for sin [A—B) its value, 

_ L sin B—D cos B 

sin C 

In like manner we have, 

D co^ A — L sin A 



(4) 



y = 



sin G 



(5) 



By taking into account the signs of the diSerent quantities, 
the above formulas (4) and (5) will solve any case that may 
arise, not only when the unmeasured sides are contiguous, as in 
the figure, but any two sides of the polygon, e,g., sides FG 
and ON, provided only that the courses are not parallel, or 
nearly so, as FG and MN. 

"'The following table gives the proper signs in a convenient 
form for reference : 



Signs of L. and D. 


Signs of functions of A. and B. 


Lat. N. ; L. is 4-- 


KE. bearing; sin 4-, cos +. 


Dep. E.; D. is +• 


S.E. bearing; sin +, cos — . 


Lat. S. ; L. is — . 


S.W. bearing; sin — , cos — . 


Dep. W.; D. is-. 


N.W. bearing; sin — , cos -f-. 



To find value of C. 

C = {A — B) if courses are in same or in opposite quadrants. 

C := {A-\-B) a the courses are in adjacent north or adjacent 
south quadrants. 

(7= 180° — (^+^) if courses are in adjacent east quadranti 
or adjacent west quadrants. 



SEC. IV.] 



AREA OK CONTEi^TS OF GEOUKD. 



119 



EXA M PL E 



Sta. 

A 


Bearings. 


Dist. 


Latitude. 


Departure. 


N. S. 


E. 


w. 


East. 


48 cb. 






48.00 




B 


K. 71i° E. 


5.18 


1.64 




4.91 




C 


S. 13i° E. 


34. 




33.13 


7.65 




D 


South. 


12. 




12.00 






E 


S. 89^ 10' W. (A) 


(x) 










F 


N. 80° 20' E. (B) 


(y) 










1.64 


45.13 


60.56 



L = +43.49 I) = —60.56 

From the above table we find that, in order to balance the 
latitude and departure columns, the total latitude of the two 
unknown courses must be 43.49 North, and the total departure 
60.56 West. Substituting in the formulas, 

43. 49 X sin 80° 20' + 60. 56 x cos 80° 20' 



X = 



y = 



sin 8° 50' 
60.56 X cos 89° 10' + 43. 49 x sin 89° 10' 



Solving, 



sin 8° 50' 
a: = 345.41; y = 288.91. 

As a check, the latitudes and departures of the two sides 
should be computed, and the survey closed as below. 









Latitude. 


Departure. 


Stations. 


Bearings. 


Dist. 








N. 


s. 


E. 


w. 


A 


East. 


48. 






48.00 




B 


N. 711° E. 


5.18 


1.64 




4.91 




C 


S. 131° E. 


34. 




33.13 


7.65 




D 


South. 


12. 




12.00 






E 


S. 89° 10' W. 


345.41 




5.02 




345.37 


F 


N. 80° 20' E. 


288.91 


48.51 




284.81 




50.15 


50.15 


345.37 


345.37 



120 ELEMEXTS OF SUEVEYING. [BOOK III. 

The following are the field-notes of a survey : 



Stations. 


Bearings. 


Distances. 


A 


North. 


7.81 ch. 


B 


S. 76^° E. 




C 


S. 10° 47' W. 


28.42 


D 


N. 84i° W. 


27.12 


E 


N. 4^° W. 




F 


East. 


16.68 



Required the distance from B to C and from E to F. 

Bio G, 17.87; ^toi^, 21.82. 



148. Another Method of Determining Areas.* 

The area of any right-line figure may be decomposed into a 
series of right trapezoids and triangles by letting fall from the 
vertices perpendiculars to any fixed straight line, called the 
Base. 

The triangles may be considered trapezoids having one 
parallel side reduced to zero. Some of the trapezoids may be 
suMradive ; but we may say that the algebraic sum of the 
trapezoids composing the series is the area of the figure. If the 
figure be bounded by a continuous or a broken curve, its area is, 
approximately, equal to that of a right-line figure whose vertices 
are contained in the curve. By properly locating these vertices. 
we may approximate the area as closely as may be required. 

Area of a Series of Right Trapezoids. 

Before deducing the necessary rules, a few preliminary defini- 
tions will be given. 

* J. Woodbridge Davis, C.E., Ph.D., 1879. 



3EC. IV.] 



AREA OR CONTENTS OF GROUND. 



121 



The position of points in a plane may be fixed by referring 
them to two intersecting straight lines of the plane. The two 
intersecting lines are called Coordinate Axes. 

Thus, let XX' and YY' 



Y' 



R 



X 



A' 



Y 

Pig. 78. 



x 



intersect at A ; the position 
of the point P is known when 
RP, its distance from YY' on 
a line parallel to XX' , and 
QP, its distance from XX on 
a line parallel to YY', are 
known. RP, or its equal 
AQ, is called the Abscissa 
of the point P; and QP, or its equal AR, is called the 
Ordinate of P. AQ and QP together are called the Co- 
ordinates of P. The point A, in which XX' and YY' 
intersect, is called the Origin of- Coordinates. When XX' 
and YY' are perpendicular to each other, as is most conyenieni, 
the coordinates of points referred to them are called Rectangu- 
lar Coordinates. 

Consider any series of five 
right trapezoids lying consecutive, 
with their bases on one straight 
Hue. The area of the series, 
as found by ordinary method, 
the symbols of Fig. 79 used, is 



Di 



D, 



d 



y 



Ds 



'D, 



D 
Fig. 79. 



i[Z),(o+J) + (A-^,)(J+c) + (A-A)(c+(i) + (A-A) 

{d + e) + {D-D,){e+f)-\. (1) 

This may be transformed into 

)^[D,{a-c)+D,{h-d)+D,{c-e)-\-D,{d-f)-^D(e-\-f)\ (2) 



From this is derived the following rule, which is true, at least, 



122 ELEMEis^TS OF SURVEYING. [bOOK III. 

for any series of five right trapezoids arranged as in Fig. 79. 
For convenience let this be called Eule A. 

Rule A. — To find the area bounded partly by any broken 
line, determined from a base line by means of rectangular 
ordinates, and otherwise bounded by the base line and terminal 
ordinates : 

Multiply the distance of each interjyiediate ordinate 
from the first by the difference between the two adjacent 
ordinates, always subtracting the following from the pre- 
ceding in order along the brohen line. Also, multiply 
distance of last ordinate from first by the sum of last 
two ordinates. Divide the sum of these products by 2. 

If this rule apply to a series of n trapezoids, its area is 

i [A {a-c)^D, (&-.^)+etc. + i)„_, {i-l)J^B^ (i + z^O]- (3) 

Add another trapezoid to the latter end of this series. Its 
area is 

i(i)„+i-i>„)(^-hO- (4) 

Add this to (3). The result is the area of a series of n-{-\ 
trapezoids, and is expressed by the following : 

|[A(f^-c) + etc.+A_i(2:->^)+i)„O*-0 + A+i(^ + 0- 

Therefore, the rule applies to a series of n -f 1 trapezoids. 

If from the series of n trapezoids the last trapezoid, whose 
area is 

i(z>„-i>„.,)(y+^). 

be subtracted, the area of the remaining series of w — 1 trapezoids 
is the following : 

J [A (a— 6)^1), (J_6?)+etc. + Z)„_i (/+./)]. 
Therefore, the rule applies to a series oin—X trapezoids. 



SEC. IV.] 



AREA OR CONTENTS OF GROUND. 



123 



It is certain that the rule applies to a series of five trapezoids. 
Let n = 5. Then n-\-l = 6, and n — 1 = 4. Consequently, the 
rule applies to a series of six trapezoids, and to a series of four. 
For this reason it applies to series of three and seven trapezoids, 
and so on. Because a series containing any number of trape- 
Boids may be found by continually adding, or subtracting, one 
trapezoid to, or from, a series of five, and because the application 
of the rule is not affected by each such operation, it follows 
that Rule A applies to a series consisting of any number of 
trapezoids. 

If that portion of the broken line which forms the upper side 
of the last trapezoid be perpendicular to base, whether directed 
towards it or from it, the area of the last trapezoid is zero. 
But expression (4), which represents this area, is zero. There- 
fore the rule conforms to this case. If the last portion of broken 
line be retrogressive, as shown in 
Fig. 80, where the line termi- 
nates at ^3, or K^, the last 
trapezoid must be subtracted in 
order that the required area — 
htiween hroken line and ~base 
limited hy terminal ordinates — may be obtained. But in this 
case expression (4) is intrinsically negative, and therefore should, 
as before, be algebraically added to the preceding area. Hence 
the rule conforms to this case. 

The statements of last paragraph are true, whatever the 
number of trapezoids in the series; even should the trapezoid 
added compose, alone, the series. In consequence, since every 
possible broken line is some succession of progressive, retro- 
gressive, and perpendicular elements, and because the addition of 
each element and its accompanying trapezoid does not affect the 
rule, rule A, or formula (3), is perfectly general, whatever may 
be the complication of the broken line. 




Fig. 80. 



124 



ELEMENTS OF SURTEYiyG. 



[book III. 



Example. — Let it be required to determine the area of a tract 
of land whose description is as follows: 

Beginning at a poplar on the south bank of the Cumberland 
river; thence south 31 poles to a stone monument; thence east 
180 poles to a beech ; thence north 40 poles to ii sycamore on the 
bank of said river ; thence, with the meanderiugs of the same, to 
the beginning. 

It was found convenient to determine the irregular boundary 
by means of offsets from the long straight side. The positions 
and lengths of these offsets are recorded in the first two columns 
below. To obtain the area by Eule A, a third column of alternate 
differences is obtained ; and in the fourth column are placed the 
products of corresponding quantities in first and third columns: 
half the algebraic sum is the area. 



Distances. 


Offsets. 


Differences. 


Products. 





31 






15 


30 


-8 


-120 


45 


39 


— 12 


-540 


77 


42 








86 


39 


-2 


—172 


113 


44 • 


3 


339 


161 


36 


4 


644 


180 


40 


76 


13680 

1 



14663 
-832 

2 ) 13831 



Square rods 



6915J 



SEC. IV.] AREA OR COI^^TEI^TS OF GROUND. 125 

The advantages of this method of calculation are readily per- 
ceived by inspection of the above example, where every product 
except the last, is found by mental work. To illustrate this 
more forcibly, let us suppose the base line to be moved, for 
instance, 2658 poles directly south. Now, all the products 
except the last remain as simple, in fact the same, as before, 
while the last only is increased. 

The old method of calculation would now require every 
product to be increased. 

If, to find the area of an irregular figure, parallel cross 
measurements be made at appropriate places, intersecting a given 
base line at known points, the figure may be considered a series 
of trapezoids. If the base line be perpendicular to the direction 
of cross-measurements, Eule A obviously applies. If the base 
line be inclined to the cross lines. Rule A may he applied, 
but the final result must he multiplied hy the sine of the angle of 
inclination. 

EXAMPLES. 

1. Wallace Colyar agrees to sell to James Beckwith at ninety 
dollars per acre, a piece of unimproved bottom land, lying 
between the west bank of the Delaware river and the foot of the 
adjacent hill. The description of the survey is as follows : 

Beginning at a willow on the west bank of Delaware river, 
this being the northeast corner of Henry Gillespie's home place ; 
thence with the meanderings of the bank of the river — [numerous 
courses here omitted] — to a white oak, the corner of James Beck- 
with's land ; thence west with his south boundary to a spring at 
the foot of the hill ; thence meandering with the foot of the hill — 
[numerous courses omitted] — to a black walnut, Henry Gillespie's 
northwest corner ; thence, with his line east to the beginning. 
The east and west being sinuous but unchangeable natural bound- 
aries, it is only required to determine, as easily and accurately as 



126 



ELEMENTS OF SUKVEYING. 



[book III. 



possible, the area. Accordingly, a line is run due north through 
the land and cross- widths are taken at right angles. The 
distances on base line and the lengths of cross lines are recorded 
in the columns below. It is required to find the area in acres 
and the sum in dollars to be paid for the strips of land. 



Distances. 


Widths. 





38 rods. 


19 


29 


40 


32 


78 


36 


111 


51 


145 


45 


173 


40 


200 


44 


228 


50 


254 


55 


290 


58 


337 


43 



Ans. 94.05625 acres; 18465.06. 



2. Caleb Hopkins employs John Bryant to cut and burn 
the underbrush in an irregular piece of hilly woodland, at 
one dollar and a half per acre, the agreement being that 
the chain shall be allowed to rest upon the surface through- 
out its length, and not be stretched horizontally. 

A base line was run N. 20° W. through the land, and cross- 
measurements were taken in directions due east and west. The 
distances in base line and the lengths of cross lines are as 
follows: 



SEC. IV.] 



AREA OR CON^TEKTS OF GROUND. 



127 



Distances. 


Widths. 





rods. 


10 


19 


26 


41 


55 


49 


78 


52 


106 


61 


129 


50 


153 


55 


175 


39 


180 


45 


194 


31 


209 






To find the Area of a Polygon. 

It has been proved that Kule A is true for any series of 
trapezoids, whatever may be the complication of the broken 
hne which forms the upper sides of the right trapezoids ; there- 
fore the rule is true when the broken line returns to its point of 
beginning. But in this case the area required as the rule is the 
area enclosed by the broken line itself, or is the area of any 
polygon. Now D, the distance between first and last ordinates, 
becomes zero, causing the only term, 
which depends upon a sum of two 
ordinates for one factor, to disappear. 

To treat this case in the most gen- 
eral way, suppose B, C, D, E, F (Fig. ^-''' 
81), to be points anywhere situated 
in a plane, and referred in position by rectangular co-ordinates 
to any point A in same plane ; and suppose points to be con- 




FiG. 81. 



128 ELEMENTS OF SURVEYING. [BOOK III. 

nected in the order uamed by a broken line, ending again at B. 
Let the ordinates of the points be denoted by a, h, c, d, e, f, and 
the abscissas by a, h', c\ cV , e' , f. Connect the origin by a 
straight line with any of the points, as B. Then the area of 
the polygon ABCDEFBA, which is equal to the area of the 
polygon BCDEFB, is by Eule A, 

i [h' {a-c) + c' {h-cl) + d' (c-e) + e' (d-f) +/ (e-i) 

+ ^' (/-«)] (5) 

= i U>' (f-o) -{- c' {h-d) + d' (c - e) + e' (d - h) 

+ /(^-^)] (6) 

= -\\P{f'- o') + c {h'-d') + d (c'-e') + e {d'-n 

+ f(e'-h')]. (7) 

From expressions (6) (7) can be framed the following rule : 

KuLE B. — To find the area of any polygon whose vertices are 
fixed by rectangular co-ordinate measurements: 

Multiply the abscissa of each vertex by the difference 
between the ordinates of the two adjacent ve7i}ices ; or, 
■multiply the ordinate of each vertex by the difference 
between the abscissas of the tico adjacent vertices ; always 
mahing the subtraction in the same direction around 
the polygon. Half the sum of these products is the 
area. 

If the origin be placed at a vertex, one term vanishes which- 
ever way the rule is used. If one of the co-ordinate axes be 
passed through two vertices, two terms vanish, when the rule is 
used one way. 



SEC. IV. J 



AREA OR CONTENTS OF GROUND. 



129 



The determination of the areas of polygons is a problem of * 
frequent occurrence in all branches of engineering and many 
other professions. Eule B is invariably simpler than the 
ordinary formula for this case, and is therefore presented above 
in the most general terms. If for words, abacissa, ordinatef 
vertex, in Eule B, be substituted longitude, latitude, station, 
that rule applies technically to the case of land surveys. 

Example. — Recently a survey of a tract of one hundred and fifty 
square miles of coal land in Tennessee was made on the co-ordinate 
system ; that is, every corner of the old grants, of the subsequent 
sales out of them, and of the older adverse claims within them, 
and every station of a road or other traverse, as well as all 
important points, as coal mines, outcrops, bridges, crossroads, 
springs, villages, etc., etc., was fixed in position with respect 
to an assumed point, by means of rectangular co-ordinates 
directed due north and east. Thus, any parcel of land could 
be easily plotted, when wanted, in any of the geological or 
other maps ; while merely a glance at the table served to 
indicate its position in the district. When the area of such a 
piece was required, it was only necessary to proceed as in the 
following example: 



station. 


Total 
latitude. 


Total 
longitude. 


Difference 
between 
alternate 

longitudes. 


Double 
areas. 


A 
B 

C 
D 
E 


7087 ft. 
10020 
8181 
5012 

2873 


94851 ft. 

97403 
101369 
103155 

98604 


-T-1201 
— 6518 

-5752 
+ 2765 
+ 8304 


8511487 

— 65310360 

—47057112 

13858180 

23857392 



Area in sq. ft. = 



66140413 
2 



130 



ELEMEN^TS OF SURVEYING. 



[book III. 



If the latitudes were larger than longitudes, differences of 
latitudes should be made the factors with total longitudes. 

Note. — The total latitude of a station with respect to any 
assumed point, is the distance, measured on a nteridian through the 
assumed point, from the point to the foot of a perpendicular drawn 
from the station to the meridian ; and the total longitude is the dis- 
tance of the station from the meridian through the assumed point. 

To find the area of a polygon surveyed, from the field notes 
of the bearings and distances of its sides. 
After finding the latitude and departure of each course and 
balancing the survey, as already explained, make a column of 
total latitudes of the various stations referred to any one of them 
as origin. This selection of origin reduces one double area 
product to zero. Now, instead of calculating the total longitudes 
of all stations and finding the differences between alternate ones, 
according to Rule B in formula (7), we may obtain the same 
result in simpler manner by adding the departures of each pair 
of adjacent courses. Modified in this way the operation becomes 
governed by the following rule : 

Rule C—Jfioltiply the total latitude of each station 
bij the sum, of the departures of the two adjacent courses. 
The algebraic half sum of these products is the area. 

EXAMPLE. 



stations. 


Bearing. 


Distance. 


Latitude. 


Departure. 


Total 
Latitude. 


Adjacent 
Departures. 


Double 
Areas. 


N + 


s- 


E-i- 


W- 


A 


N 35° 00' E 


2.70 


2.21 




1.55 










B 


N 83° 30^ E 


1.29 


.15 




1.28 




2.21 


2.83 


6.2543 


C 


S 57" 00' E 


2.22 




1.21 


1.86 




2.36 


3.14 


7.4104 


D 


S 34° 15' W 


3.55 




2.93 




2.00 


1.15 


-0.14 


-0.1610 


E 


N 56° SC W 


3.23 


1.78 






2.69 


-1.78 


-4.69 


8.3482 



Square chains 



2 )21.8519 
. 10.9259 



SEC. v.] MAGN'ETIC DECLINATIOIf. 131 

To find the total latitude of each station, add to total latitude 
of preceding station the latitude of preceding course. If the 
latitude of last station, found in this way, be equal to the 
latitude of last course with reversed sign, the work is correct. 
However, the latitude of first station, or station taken as origin, 
is always zero ; of last station it is always the latitude of last 
course with reversed sign ; and of second station it is always the 
latitude of first course. To find the adjacent departures^ add the 
departures of the two courses, one on each side of the station. 



SECTION V. 

MAGNETIC DECLINATION OR VARIATION OF THE 

NEEDLE. 

The following articles, 149-164, are essentially, and so far 
as practicable verbatim, from papers of the U. S. Coast and 
Geodetic Survey. 

149. The magnetic declination at any place is the angle 
which the compass-needle, when it is correctly constructed and 
freely suspended, makes with the true meridian. The true 
meridian is fixed, but the declination varies because the direc- 
tion in which the needle points is in a continuous state of 
change. Therefore, whenever a measure of the declination of 
the needle is taken, the exact time (year, day of month, and 
hour of the observations) should be recorded as well as the 
geographical position of the place, or its latitude and longitude 
expressed to the nearest minutes of arc. 

150. The declination is called ^'West" when the north end 
of the needle points to the west of the true meridian, and it is 



132 ELEMENTS OF SURVEYIN'G. [BOOK 111. 

called ^^East" when the north end of the needle points east of 
the true meridian. In order to give an idea of the amount of 
the declination at present observable within the limits of the 
United States we instance the followiDg places at or near which 
it reaches extreme value, which are given to the nearest whole 
degree (1878) : 

At Eastport, Me., decUnation 18° west. 

At the mouth of the Eio Grande, Texas, 8° east. 

At San Diego. Cal., 14° east. 

At Sitka, Alaska, 29° east. 

At Fort Yukon, Alaska, 36° east. 

151. The accuracy with which the declination may be deter- 
i£iined depends chiefly upon the instrumental means, but also, 
and in a great measure, upon the care taken in the use of the 
instruments and the selection of the proper methods and times 
for observing. 

Omitting any detailed notice of the irregular variations to 
which the magnetic needle is subject, it becomes important for 
the purposes of the surveyor to refer particularly to the changes 
which have a special bearing upon his observations. These are 
the daily variation and the secular variation. 

152. The Daily Variation. —It has been found that at 
about the time of sunrise the north end of the needle has a slow 
motion towards the east which soon ceases. The needle is then 
said to be at its eastern elongation ; its north end then begins a 
retrograde motion towards the west, and at about one o'clock 
in the afternoon reaches the point at which it is said to be at 
its western elongation, after which it again turns back towards 
the east. 

The times at which the needle reaches its eastern and 
western elongations vary with the seasons of the year (with 



BEC. v.] 



MAGI^'ETIC DECLIN^ATIOi?'. 



133 



the sun's declination), happening a little earlier in summer 
than in winter. 

The angular range between the eastern and western elonga- 
tions varies also with the season of the year. 

The average position of the needle for the day is called the 
mean magnetic meridian. 

At about six o'clock in the evening (and for about an hour 
before and after), throughout the year, the position of the needle 
coincides very nearly with the mean magnetic meridian, and this, 
therefore, is the time most favorable for making observations to 
obtain at once the mean declination. 

153. For reducing the direction of the needle observed at 
other hours to the mean magnetic meridian the following table is 
furnished. It gives to the nearest minute the variations of the 
needle from its average position during the day, for each hour 
in the day for the four seasons of the year. 

Table for reducing the observed declination to the mean decli- 
nation of the day. 





The needle points east of the 


The needle points west of the mean magnetic 




mean magnetic meridian. 






meridian. 




A.M. 


A.M. 


A.M. 


A.M. 


A.M. 


A.M. 


a 

O 

o 


P.M. 


P.M. 


P.M. 


P.M. 


P.M. 


P.M. 
















^ 
















h. 


h. 


h. 


h. 


h. 


h. 


§ 


h. 


h. 


k. 


h. 


h. 


h. 


Bour 


6 


7 


8 


9 


10 


11 


^ 


1 


2 


3 


4 


5 


6 




/ 


/ 


/ 


1 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


Spring 


3 


4 


4 


3 


1 


1 


4 


5 


5 


4 


3 


2 


1 


Summer.... 


4 


5 


5 


4 


1 


2 


4 


6 


5 


4 


3 


2 


1 


Autumn .... 


2 


3 


3 


2 





2 


3 


4 


3 


2 


1 


1 





Winter 


1 


1 


2 


2 


1 





2 


3 


3 


2 


1 


1 






154. It appears from observations of the daily fluctuation of 
the declination that the mean of the extreme easterly and 



134 ELEMENTS OF SURVEYING. [BOOK III. 

westerly positions m any one day approaches nearly (within half 
a minute) to the mean position of the day, as derived from 
hourly observations continued day and night. Since corrections 
to observed declinations to refer them to the mean of the day are 
generally very unsatisfactory, it is recommended to observe the 
declination for any one day at the epochs of the eastern magnetic 
elongation and of the western magnetic elongation and to take 
the mean position as representing the declination for that day. 
The epochs of extreme positions, as observed at Philadelphia, 
Washington, and Key West, apply, with comparatively small 
changes, to nearly all places within the United States and may 
be stated to be as follows : Referring to the north end of the 
magnet, the morning eastern elongation occurs, on the average, 
from May to September, inclusive, about 7-|- a.m.; in March, 
April, and October, about 8 a.m.; in November, about 8|- a.m., 
and in December, January, and February, about 9 a.m.; earliest 
time in August, about 7Ja.m. ; latest in January, about 9 a.m. 
These epochs, however, are subject to great fluctuations and 
cannot be depended upon in any one case within one hour and 
frequently they cannot be recognized at all, either on account of 
the small range of the daily fluctuation — the amount of which in 
winter is but one-half, nearly, of the amount in summer — which 
is easily disguised by small irregularities, or on account of 
disturbances, which reach their maxima in September and 
October, and generally are more predominant in winter than in 
summer. The afternoon loestern elongation occurs, on the 
average, about IJ p.m. from May to November, inclusive, and 
about 1| P.M. in the remaining months; also, earliest in Septem- 
ber — some minutes before 1 p.m. — and latest in January, about 
If- P.M. The afternoon epoch is subject to less fluctuation than 
the morning epoch. 

155. The Secular Variation of the magnetic declination 



SEC. Y.] MAGNETIC DECLIITATIO]^". 135 

is a subject of the greatest importance to surveyors. It mani- 
fests itself by a gradual change in one direction, which at first 
increases slowly, then more rapidly, diminishing again after- 
ward until the needle becomes stationary and subsequently 
returns by similar changes to its former position, the whole 
period extending over nearly two and a half centuries. Thus, 
at Philadelphia the declination was 8f° west in 1700, whence it 
diminished until in 1800 it reached a minimum 2°.l (2° 6'), 
and increased again to 6°.8 in 1880. At present all along 
the Atlantic and Gulf coasts the effect of the secular variation 
is to i7icrease west declinations or to decrease east declinations 
by from 2' to 5', but on the Pacific coast the effect is opposite 
in direction, viz., increasing east declinations by from 1' to 3'. 
In Alaska, however, there are indications of a decrease of east 
declination. 

156. Lying between the region in which the variation of 
the needle is west and that in which such variation is east, 
is a line of no variation, on which the magnetic meridian 
coincides with the true meridian. 

The north (or seeking) end of the needle always inclines 
toward the line of no variation : hence, for all points east of 
this line, the variation is West; and for all points west of it, 
the variation is East. 

This line is not a fixed line, but changes its position from 
year to year. By referring to a map, such line for the United 
States may be traced. In 1870, it crossed Sault St. Marie at the 
lower end of Lake Superior, passed, very nearly, through Cleve- 
land in the State of Ohio, Ealeigh in North Carolina, and 
passed into the Atlantic Ocean near Wilmington, North Caro- 
lina. In 1875, it crossed Lake Superior and entered Michigan 
at White Fish Point, passed thence, very nearly, through Bay 
City Michigan, Oberlin Ohio, Parkersburgh West Virginia, 



136 



ELEMENTS OF SURVEYIl^^G. 



[book III. 



thence, with a slight curve to the southwest, to Fajetteville 
North Carolina, and thence into the Atlantic Ocean a little 
to the west of Cape Fear. The line of no variation is now 
moving quite rapidly to the south and west, and therefore it 
is that ivest declinations of the needle are increasing, and east 
declinations decreasing. 

157. The U. S. Coast and Geodetic Survey has made a 
collection of magnetic declinations observed at various places in 
the United States and elsewhere. Mr. Charles A. Schott, chief 
of the Computing Division of the Survey, has tabulated the 
observed declinations, and deduced from them a Table of For- 
mulae. This table is here given so far as it relates to localities 
in the United States. The localities are arranged geographically, 
as far as practicable, and their positions are given by latitude and 
longitude (west of Greenwich). 



Formulae expressing the magnetic declination at various places 

in the United States. 



Locality. 


Latitude. 


Longitude. 


Expression for Magnetic Declination. 


Portland, Me 


o 

43 


r 
38.8 


o 

70 


16.6 


O 

D^ +10.72 + 2.68 sin (1.33 ?w+ 24.1). 


/ Burlington, Vt 


44 


28.2 


73 


12.3 


D= +10.81 + 3.65 sin (1.30 w— 20.5) 

+ 0.18 sin (7.0 m+132). j 


Rutland, Vt 


43 


36.5 


72 


55.5 


D= f 10.03 + 3.82 sin (1.5 m— 24.3). 


Portsmouth, N. H 


43 


04.8 


70 


43.0 


D= +10.63 + 3.17 sin (1.44 m— 4.7). 


1 Newburyport, Mass.. 


42 


48.4 


70 


49.0 


D= +10.07 + 3.10 sin (1.4 m+ 1.9). 


Salem, Mass 


42 

42 


31.9 
21.5 


70 
71 


52.5 
03.8 


D=+ 9.80 + 3.61 sin (1.50 m— 1.0). 
D=+ 9.52 + 2.93 sin (1.30 7n+ 5.0). 


Boston, Mass ... 


Cambridge. Mass 


42 


22.9 


71 


07.7 


D=+ 9.58 + 2.69 sin (1.3 m+ 7.0) 
+ 0.18 sin (3.2 W2+ 44). 


Nantucket, Mass. . . . 


41 


17.0 


70 


06.0 


D=+ 9.29 + 2.78 sin (1.35 OT+ 5.5). 


Providence, R. I 


41 


49.5 


71 


24.1 


D=+ 9.10 + 2.99 sin (1.45 m— 3.4) 
+ 0.19 sin (7.2 m + 116). 


Hartford, Conn 


41 


45.9 


72 


40.4 


D=+ 8.06 + 2.90 sin (1.25 m— 26.4). 


Wew Haven, Conn . . . 


41 


18.5 


72 


55.7 


D=+ 7.78 + 3.11 sin (1.40 m— 22.1). 



SEC. v.] 



MAGNETIC DECLIiq^ATIOJf. 



13? 



Locality. 


Latitude. 


Longitude. 


Expression for Magnet.^ Oeclination. 


Albany, N.Y 


42 


39.2 


73 


45.8 


D=+ 8.17 + 3.02 sin (1.4im— 8.3). 


Oxford, N. Y 


42 
42 


26.5 
52.8 


75 

78 


40.5 
53.5 


D=+ 6.19 + 3.24 sin (1.35 7?i— 18.9). 
D=+ 3.66+3.47 sin (1.4 w— 27.8). 


Buffalo, N.Y 


Erie, Pa.. 


42 


07.8 


80 


05.4 


D=+ 2.26 + 2.71 sin(l 55 m— 29.7). 


Cleveland, Ohio 


41 


30.3 


81 


42.0 


D=+ 0.10 + 2 07 sin (1.40 7n— 6.2). 


Detroit, Mich 


42 


20.0 


83 


03.0 


D=— 0.97 + 2.21 sin (1.50 m— 15.3). 


Saint Louis, Mo 


38 


38.0 


90 


12.2 


D=— 7.15 + 2.33 sin (1.4 m— 20.1).* 


New York, N.Y 


40 


42.7 


74 


00.0 


D=+ 6.40 + 2.29 sin (1.6 m— 5.5) 
+ 0.14 sin (6.3 m+ 64). 


Hatborough, Pa 


40 


12 


75 


07 


D=+ 5.23 + 3.28 sin (1.54 771^ 13.2) 
+ 0.22 sin (4.1 wi+157). 


Philadelphia, Pa 


39 


56.9 


75 


09.0 


D=+ 5.38 + 3.29 sin (1.55 m— 23.9) 
+ 0.39 sin (4.0 m + 161). 


Harrisburg, Pa 


40 


15.9 


76 


52.9 


D=+ 2.93 + 2.98 sin (1.50 m+ 0.2). 


Baltimore, Md 


39 


17.8 


76 


37.0 


D=+ 3.20 + 2.57 sin (1.45 m— 21.2). 


Washington, D. C 


38 


53.3 


77 


00.6 


D=+ 2.47 + 2.47 sin (1.40m— 14.6). 


Cape Henry, Va 


36 


55.5 


76 


00.5 


D=+ 2.54 + 2.41 sin (1.50 m— 35.4). 


Charleston, S. C 


32 


46.6 


79 


55.8 


D=- 2.14 + 2.74 sin (1.35 m 1.3). 


Savannah, Ga 


32 


04.9 


81 


05.5 


D^- 2.54+2.32 sin (1.5 m- 28.6). 


Key West, Fla 


24 


33.5 


81 


48.5 


D=— 3.90 + 2.93 sin (1.4 m— 33.5). 


Mobile, Ala 


30 


41.4 


88 


02 5 


D=— 4.40 + 2.69 sin (1.45 m— 76.4). 
D=— 5.61 + 2.57 sin (1.4 m- 61.9). 


New Orleans, La 


29 


57.2 


90 


03.9 


San Diego, Cal 


32 


42.1 


117 


14.3 


D=— 12. 54 + 1. 64 sin (1.2 m— 180.0). 


Monterey, Cal 


36 


36.1 


121 


53.6 


D=— 12. 82 + 3.54 sin (1.0 m-142.9). 


San Francisco, Cal. . . 


37 


47.5 


122 


27.2 


D=— 13.34 + 3.23 sin (1.00 m— 130.3). 


Cape Disappointment, 












Wash. Ter 


46 


16.7 


124 


02.0 


D=— 20. 72 + 2. 81 sin (1.2 m— 188.8). 


Sitka, Alaska 


57 


02.9 


135 


19.7 


D=— 26.72 + 2.41 sin (1.6 m-107.1). 


Unalashka, Alaska. . . 


53 


52.6 


166 


31.5 


D=— 18. 34 + 1.45 sin (1.4 m— 67.8). 



158. The epoch to which the formulae refer is 1850 ; hence, 
in the "expression for magnetic decHnation/' 

m — t — 1850. 
To illustrate the use of the table. Let it be required to find the 
declination of the needle at Albany, N. Y., in August, 1879, 
or 1879.6. 



* Approximate expreBsion, 



138 ELEMEi^TS or SUKVEYIls^G. [BOOK lil 

The tabular expression for magnetic declination at Albany is 
D = +8M7 + 3^02 sin (1.44° w-8°.3). 
m = 1879.6 — 1850 = 29.6. 
1.44° m — 42°.624. 

.-. 1.44°m-8°.3 = 34°.324. 
Natural sin (34°.324) = .5638 ; 

hence, -Z> = + 8°.17 + 3°.02 x .5638 = + 9°.87 ; 

which gives the computed declination of the needle at Albany in 
1879.6, as 9°.87 (west, since the result is phcs), which differs from 
the observed declination at that time but one-hundredth of a 
degree. 

If the time for which the declination is desired is prior to 
1850, then m will be negative. For example, let the declination 
at Albany for 1836.8 be required ; then 

m = 1836.8 — 1850 = —13.2, 

and 1°.44 m = 1°.44 x ( — 13.2) = -19°.008, 

and 1°.44 ???- 8°.3 = — 27°.308, 

and sin (— 27°.308) = -.45877; 

.-. D = 8M7 + 3°.02 sin (— 27°.308) = 8°.17-1°.39 = +6°.78; 

which agrees exactly with the observed declination. 

The student is referred to Mr. Schott's paper. Appendix 
No. 9, IT. S. Coast and Geodetic Survey Report for 1879, for 
valuable tables of Magnetic Declinations. 

159. From the same paper is taken the following table of 
computed annual changes in the declination of the magnetic 
needle for 1870, 1880, and 1885, expressed in minutes of arc, a 
+ sign indicating north end of needle moving westward, a — 
sign indicating north end moving eastward: 



SEC. v.] 



MAGNETIC DECLINATION?". 



139 



TABLE. 



Locality. 



Annual change. 



In 1870. 



In 1880. 



In 1885. 



Portland, Me.. 

Burlington, Vt 

Eutland, Vt 

Portsmouth, N. H. 
Newburyport, Mass 

Salem, Mass 

Boston, Mass 

Cambridge, Mass. . . 
Nantucket, Mass. . . 

Providence, E. I 

Hartford, Conn. . . . 
New Haven, Conn. 

Albany, N.Y 

Oxford, N. Y 

Buffalo, N. Y 

Erie, Pa 

Cleveland, Ohio. . . . 

Detroit, Mich 

Saint Louis, Mo. . . . 
New York, N. Y. . . 
Hatborough, Pa. . . 
Philadelphia, Pa. . . 

Harrisburg, Pa 

Baltimore, Md 

Washington, D. C. 

Cape Henry, Va 

Ch.^rleston, S. C . . . 



+ 2.4 
+ 5.0 
+ 6.0 
+ 4.4 
+ 3.9 
+ 5.0 
+ 3.4 
+ 2.9 
+ 3.3 
+ 3.8 
+ 3.8 
+ 4.6 
+ 4.3 
+ 4.5 
+ 5.1 
+ 4.4 
+ 2.8 
+ 3.4 
+ 3.4 
•+2.4 
+ 4.6 
+ 4.9 
+ 4.1 
+ 3.9 
+ 3.5 
+ 3.8 
+ 3.5 



+ 1.6 
+ 6.0 
+ 5.6 
+ 3.7 
+ 3.3 
+ 4.1 
+ 2.9 
+ 2.1 
+ 2.7 

+ 3.7 
+ 4.3 

+ 3.7 
+ 4.3 
+ 5.0 
+ 4.2 
+ 2.5 
+ 3.0 
+ 3.2 
+ 2.5 
+ 4.5 
-i-4.9 
+ 3.3 
+ 3.6 
+ 3.2 
+ 3.7 
+ 3.0 



+ 1.2 
+ 5.8 
+ 5.3 
+ 3.3 

+ 2.9 
+ 3.5 

+ 2.5 
+ 1.8 

+ 2.4 

+ 3.6 
+ 4.1 
+ 3.4 

+ 4.0 
+ 4.8 
+ 4.0 
+ 2.2 
+ 2.8 
+ 3.0 
+ 2.6 

+ 5.3 
+ 2.8 
+ 3.2 
+ 3.0 
+ 3.6 
+ 2.7 



140 



ELEMENTS OF SURVEYING. 



[book III. 



Locality. 



Annual change. 



In 1870. 



lu 1880. 



In 1885. 



Savannah, Ga 

Key West, Fla 

Mobile, Ala 

New Orleans, La 

San Diego, Oal 

Monterey, Gal 

San Francisco, Gal 

Gape Disappointment, W. Ter. 

Sitka, Alaska . . . 

Unalashka, Alaska 



+ 3.6 
+ 4.3 
+ 2.8 
+ 3.1 

— 1.9 
-2.0 

— 1.0 
-3.4 
+ 1.0 
+ 1.6 



+ 3.5 

+ 4.2 
+ 3.4 
+ 3.5 
-1.7 
—1.5 
—0.5 
-3.1 
+ 2.1 
+ 1.9 



+ 3.3 
+ 4.1 

+ 3.7 
+ 3.7 
-1.6 
— 1.1 
-0.3 
-2.7 
+ 2.5 
+ 2.0 



160. It will be observed that the amount of change is by no 
means the same, even in places not far remote from each other, 
as New York and Philadelphia. 

In grouping together a table of the present rate of change, 
much allowance must therefore be made for possible local pecu- 
liarities that have not been ascertained. 

A surveyor should, at the time and place of survey, deter- 
mine the true meridian and thence the magnetic declination for 
record with his survey. 



Method of ascertaining the Declination. 

161. The best practical method of determining the true 
meridian of a place, is by observing the north star, Polaris. 
If this star were precisely at the point in which the axis of the 
earth prolonged pierces the heavens, then the intersection of 
the vertical plane passing through it and the place, with the 
surface of the earth, would be the true meridian. But the 



SEC. v.] MAGNETIC DECLINATION". 141 

star being at a distance from the pole equal to 1° 30' nearly, 
it performs a revolution about the pole in a circle, the polar 
distance of which is 1° 30' nearly, and the time of revolution is 
23 hours and 56 minutes. 

To the eye of an observer this star is continually in motion, 
and is due north but twice in 23 hours and 56 minutes, and is 
then said to be on the meridian. When it departs from the 
meridian, it apparently moves east or west for 5 hours and 
59 minutes, and then returns to the meridian again. When at 
its greatest distance from the meridian, east or west, it is said to 
be at its greatest eastern or western elongation. 

162. The western elongations from the beginning of April to 
the end of September, and the eastern from the beginning of 
October to the end of March occur in the day-time. If it be 
necessary to determine the meridian at that particular season of 
the year, and at night, let 5 hours and 59 minutes be added to or 
subtracted from the time of greatest eastern or western elongation, 
and the observation be made at night when the star is on the 
meridian. 

163. The angle which the meridian plane makes with the 
vertical plane passing through the pole-star when at its greatest 
eastern or western elongation, is called the Azimuth of Polaris. 

The following extract, Art. 164, with the tables, is from the 
Report of the U. S. Coast and Geodetic Survey for 1881 : 

164. The following tables of the times and azimuths of Po' 
laris ivhen at elongation have been prepared for the benefit of those 
surveyors and others who may prefer to make use of the pole-star 
for their determination of the true meridian, and whose instru- 
mental outfit for the measure of the declination may be limited to 
a compass -svith sights or to a small theodolite with compass-needle 
attached, and who may be without a chronometer. 



143 



ELEMENTS OF SURVEYIi^^G. 



[book III. 



The method was recommended to surveyors by Dr. Charles 
Davies in the revised edition of his work on surveying, and a 
description of it still forms part of the instructions of the Com- 
missioner of the General Land Office to the surveyors-general of 
public land of the United States (editions of 1855, 1871, and 1878). 
The tables given in these instructions have either become obsolete 
from lapse of time or are not sufficiently extended for future use. 
They were, therefore, recomputed, and in their present form and 
with the rules given for interpolation will be found to possess 
greater accuracy than any similar tables previously published. 
The tables include all elongations whether occurring by day or 
night. Polaris may be observed in day-time w^hen the sun is not 
too high even with moderately powerful telescopes ; besides, a 
complete table facilitates interpolation. 

Mean local time (astronomical, counting from noon) of the 

elongations of Polaris. 

[The table answers directly for the year 1885, for latitude +40° and for longitude 6 hours 

west of Greenwich.] 



Date. 


Eastern 


elongation. 


Western elongation. 


Jan. 


1 


h. 



m. 

35.3 


h. 
12 


m. 

24.6 


(< 


15 


23 


36.1 


11 


29.3 


Feb. 


1 


22 


29.0 


10 


22.2 


iC 


15 


21 


33.7 


9 


27.0 


Mar. 


1 


20 


38.5 


8 


31.8 


(< 


15 


19 


43.4 


7 


36.6 


Apr. 


1 


18 


36.4 


6 


29.7 


a 


15 


17 


41.4 


5 


34.7 


May 


1 


16 


38.6 


4 


31.8 


i< 


15 


15 


43-7 


3 


36.9 


June 


1 


14 


37.1 


2 


30.3 


iC 


15 


13 


42.2 


1 


35.4 



SEC. v.] 



MAGNETIC DECLIiq-ATIGlC. 



143 




Date. 


Eastern elongation. 


Western elongation. 


July 


1 


h. 


m. 

39 6 


h. 



m. 

32.8 


a 


15 


11 


44.7 


23 


34.0 


Aug. 


1 


10 


38.2 


22 


27.5 


i( 


15 


9 


43.3 


21 


32.6 


Sept. 


1 


8 


36.7 


20 


26.0 


a 


15 


7 


41.7 


19 


31.1 


Oct. 


1 


6 


38.9 


18 


28.2 


a 


15 


5 


43.9 


17 


33.2 


Nov. 


1 


4 


37.0 


16 


26.4 


i( 


15 


3 


41.9 


15 


31.3 


Dec. 


1 


2 


38.9 


14 


28.2 


i( 


15 


1 


43.6 


13 


33.0 



N. B. — To refer the tabular times to any year (limit about 
10 years) subsequent to the epoch, add 0"\35 for every year. For 
years previous to epoch subtract 0°'.35 for every year. 

To refer the tabular times to any other latitude (between the 
hmits 25° and 50°), add 0'".14 for every degree south of 40° ; 
subtract 0°'.18 for every degree north of 40°. 

To refer the tabular times to any year in a quadriennium, 

For first year after a leap year the table is perfect. 

For second year after a leap year add . . . I'^.O 

For third year after a leap year add . . . 2"*. 

For a leap year and before March 1 add . . d'^.O 

And for remainder of the year subtract . . 1™.0 

For any other than the tabular day subtract from the tabular 
time of elongation 3"'.94 for every day elapsed. 

It will be noticed that there occur two eastern elongations on 
January 9, and two western elongations on July 9. 



144 



ELEMENTS OF SUEVEYING. 



[book III. 



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r— « 









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SEC. v.] MAGNETIC DECLINATIOif. 145 

To find the true Meridian with the Compass. 

165. 1. Drive two posts firmly into the ground, in a line 
nearly east and west ; the uppermost ends, after the posts are 
driven, being about three feet above the surface and at the same 
level, and the posts about four feet apart ; then lay a plank, or 
piece of timber three or four inches in width, and smooth on 
the upper side, upon the posts, and let it be pinned or nailed, 
to hold it firmly. 

2. Prepare a piece of board four or five inches square, and 
smooth on the under side. Let one of the compass-sights, or a 
piece of tin with a very small perforation, be fastened at right 
angles to the upper surface of the board, and place a brick or 
other weight upon the ^sight-board to keep it steady. 

3. At about twelve feet from the stakes, and in the direction 
of the pole-star, let a plumb be suspended from the top of an 
inclined stake or pole. The top of the pole should be of such 
a height that the pole-star will appear about six inches below it ; 
and the plumb should be swung in a vessel of water to prevent 
it from vibrating. 

This being done, about twenty minutes before the time of 
elongation, place the board, to which the compass-sight is fastened, 
on the horizontal plank, and slide it east or west, until the 
aperture of the compass-sight, the plumb-line, and the star, are 
brought into the same range. Then if the star depart from 
the plumb-line, move the compass-sight, east or west, along 
the timber, as the case may be, until the star shall attain its 
greatest elongation, when it will continue behind the plumb-line 
for several minutes ; and will then recede from it in the direc- 
tion contrary to its motion before it became stationary. Dur- 
ing this observation it will be necessary to have the plumb- 
line lighted ; this may be done by an assistant holding a candle 
near it. 



146 ELEMEifTS OE SURVEYING. [BOOK J II. 

'Now let a point be fixed by a second plumb-line placed at a 
distance of thirty or forty yards from the first, and in the same 
direction with it and the compass-sight. Mark tlie points 
determined by the two plumbs, for future reference, by driving 
pegs deeply into the ground and marking the exact point on each 
by a small nail. The line so determined, makes, with the true 
meridian, an angle equal to the azimuth of the pole-star ; and, 
from this line, the yariation of the needle is readily determined, 
even without tracing the true meridian on the ground. 

Place the compass upon this line, turn the sights in the 
direction of it, and note the angle shown by the needle. Now, 
if the elongation at the time of observation was west, and the 
north end of the needle is on the west side of the line, the 
azimuth plus the angle shown by the needle, is the true varia- 
tion. But should the north end of the needle be found on the 
east side of the line, the elongation being west, the difference 
between the azimuth and the angle would show the variation ; 
and the reverse when the elongation is east. 

It may be stated that for magnetic purposes a moderate 
degree of accuracy suffices in the determination of the true 
meridian, and a correct knowledge of it within 1' will in general 
fully suffice. It is difficult, even in our middle latitudes, to 
determine the magnetic meridian within the limit of 1' on 
account of the continuous fluctuations, hence any greater accu- 
racy than this in the astronomic meridian would be useless. 

166. A very near approximation to a true meridian, and- 
consequently to the variation, may be had, by remembering 
that the pole-star very nearly reaches the true meridian, when 
it is in the same vertical plane with the star Delta (6) in the 
constellation Cassiopeia. 

Using the same apparatus as in Art. 165, place the "sight- 
board" in line with the plumb-line and the pole-star, and 



SEC. v.] 



MAGNETIC DECLINATION. 



147 



JiVdQ 



• S 









x> 



move it to the east as the pole-star moves ^vesi, till Polaris and 
Delta both appear upon the 
plumb-line together; the line 
through the point of sight and 
the plumb-line will be, very 
nearly and with sufficient accu- 
racy, the true meridian. This 
method is practicable only when 
the star Delta is 'below the 
pole-star during the night ; 
when it passes the meridian 
above the pole, it is too near 
the zenith to be of service, in 
which case the star Zeta (^), 
the last star but one in the 
tail of the Great Bear, may be 
used instead. 

Delta Cassiopeiae is on the 
meridian below the pole-star 
at midnight about April 10, and 
is, therefore, the proper star 
to use at that date and for 
some two months before and 
after. 

Six months later, the star 
Zeta in the tail of the Great 
Bear will supply its place, and 
is to be used in precisely the 
same manner. 

Fig. 82 gives a representa- 
tion, drawn to scale, of N. Pole, 
Polaris, and the Constellations 
Cassiopeia and Great Beai ; and fw. 82. 



Polaris 



•IT.Tolt 



£• 



Casd 



•/3 



a" 

-6* 



opeia 



li 



148 



ELEMENTS OF SUKVEYING. 



[book III. 



the line drawn through the star Delta (6) of Cassiopeia and Zeta 
(C) of the Great Bear, represents those stars on the meridian with 
the pole-star. 

The method given in this article for finding the true meridian 
cannot be used with advantage, on account of the haziness of 
the atmosphere near the horizon, at places below about 38° 
north latitude. 

167. The variation of the needle should always be noted on 
every survey made with the compass, and then if the land be sur- 
veyed at a future time, the old lines can always be re-run. 

In re-running the lines of an old survey, a vernier is used for 
setting off the variation of the needle. 

168. A Vernier is a contrivance for measuring snaaller divi- 
sions of a unit than those into which the line, to which it is 
applied, is divided. It is a graduated scale so arranged as to 
cover an exact number of equal spaces on the primary scale, or 
limb. It is divided into a number of equal parts greater by one 
than the number of equal spaces which it covers on the limb. 

The vernier may be applied to any limb or scale of equal parts. 
The modes of its application are extremely various ; the principle, 
howeyer, is the same in all, and may be illustrated by a simple 
diagram (Fig. 83). 



W // 12 IS 14- iJ 16 17 18 



19 



























c 


1 1 1 1 1 1 1 1 


n 





B 



6 6 
Fig. 83. 



to 



Let ^^ be any lirrib or scale of equal parts, one of which let 
us suppose equal to ^ of an inch. Let CD be a vernier, equal 
in length, say to nine of these parts, and itself divided into ten 
equal spaces, each one of which is then equal to nine-tenths of 
^^ or yfo of an inch. The difference between a space on the 
limb and a space on the vernier, is therefore equal to one- tenth of 



SEC. v.] MAGKETIC DECLINATION. 149 

^^ or y^ of an inch. This is the least space that can be 
measured by means of the vernier, and is called the least count j 
hence, 

The least count of a vernier is equal to one of the equal 
divisions of the limb divided by the number of spaces on the 
vernier. 

169. The true reading of an instrument, for any position of 
the vernier, expresses the distance from the point where the 
graduation on the limb begins, marked 0, to the point of 
the vernier. In the diagram, that distance is expressed by 
nine units of the limb, or 9. 

If, now, the vernier be moved till the division 1 coincides 
with the division 10 of the limb, the point will have advanced 
along the limb a distance equal to -^ of one space, and, if we call 
each space J, the reading will become ^-\-^b. If we again move 
the vernier till the division 2 coincides with the division 11 of 
the limb, the point will have advanced an additional distance, 
equal to -j^h, and the reading becomes ^-{-^-^h ; when 3 coincides 
with division 12, the reading will become 9 + ^^, and so on, till 
finally, when the point 10 coincides with 19 of the limb, the 
distance 9 will have been increased by \^h, and will become 10, 
as it should, since, in that case, the point will have been 
moved a whole space, and will coincide with the division 10 
of the limb. Hence, the following rule for reading an instru- 
ment which has a vernier : 

Read the scale, in the direction of the graduation, up to the 
line preceding the of the vernier ; this gives the number of 
whole units of the scale. Look along the vernier and find which 
of its lines coincides with a line of the scale ; this line of th© 
vernier gives the number of fractional parts of one unit of the 
scale to be added to the former reading. 



150 



ELEMENTS OF SUKVEYING. 



[book III. 



170. In order to read a vernier correctly, especially if it be 
one with which we are not familiar, it is necessary to estimate, 
hy the eye alone, the fractional part of one unit to be added ; 
then if the vernier reading is nearly the same as the estimate, 
we may record the reading with confidence ; but if the estimate 
and the vernier reading disagree largely, then the cause of such 
disagreement will probably be a false reading of the vernier. 



H 




-6,5^; 



a^^^^ 



^s. 



FiQ. 84. 

171. In a vernier compass, the vernier is attached, as 
shown in the figure. A small arc HI is described on the bar 
AB, having its centre at the centre of the compass-box. This 
arc is divided to degrees, and sometimes to the parts of a degree. 
The vernier is permanently attached to the compass-box. 

When the point of this vernier coincides with the point 
of the graduated *arc HI, the north and south line of the com- 
pass-box lies in the plane of the sights. 

The compass-box is turned about its centre, without moving 



SEC. v.] . MAGNETIC DECLIi^ATION^. 151 

the plate AB, by means of the milled screw L : and is fastened 
to the plate AB, by a clamping nut underneath the main 
plate. 

172. To set off the variation of the needle on a yernier 
compass, turn the north end of the compass plate by the tangent 
screw L of the vernier (see Fig. 84), over the number of degrees 
in the variation, to the left (the observer is supposed to be 
standing at the south end of the plate and looking towards 
the north end), if the variation is westerly, to the right if 
it is easterly. 

173. To run out lines of old deeds, set the compass upon 
one end of an ^^ original line," determined to be such by old 
"marks," or by testimony as to its having been undisputed 
for twenty years or more, and turn the vernier plate till the 
reading is the same as that given in the deed ; then run 
the other lines by the bearings and distances as given in 
the deed. 

174. To make allowance for change in "varia'tion" in 
running out old lines, when only one "corner" can be identi- 
fied, the surveyor must see to it that the date of the deed 
from which he is working is the same as the date of the survey 
from which the description of property in the deed is taken ; 
very often descriptions are copied from former deeds, in 
which case the date of the survey must be discovered before 
running the lines. 

175. If no *^ original" line of the property can be found, 
nor a corner and date of survey, then the surveyor must seek 
his data upon neighboring property and work back from there, 
through several deeds if necessary. 

176. It is very desirable that the date of the survey from 



152 ELEMENTS OF SURVEYIN^G. [BOOK lU. 

which a description is taken, and also the name of the surveyor, 
should be entered in every deed conveying lands. 

The form of a *^ Survey Bill '* may be somewhat as follows : 

Description of Property in Town of , County of 

, State of , surveyed on the sixteenth day of 



August, in the year eighteen hundred and eighty- three, and 
bounded as follows : 

Beginning at a point in the northwest corner of land owned 
by A. B. 0., and marked by a cross cut into the top of a stone 
sunk three feet into the ground, — from which point the east 
chimney of the house of A. B. 0. bore S 87^° E, distant 8.56 ch.— 
thence from the point of beginning N 18° E, 11.27 ch. to a 
stake ; — thence N 56° E, 15.26 ch. to a corner where two stone 
walls meet; — thence S 87° E, 5.56 ch. to the shore line of the 
bay ; — thence following the shore line to a point on the shore 
Hne bearing S 23° W, and distant in a straight line 13.47 ch. — 
from which point the middle of the eastern key-stone of the 
bridge across Beaver Dam Creek bears S 37|-° E ; — thence, fol- 
lowing the middle of the road to a point in the middle of the 
road bearing S 87° W, distant in a straight Hne 5 ch. — thence, 
&c., &c. ; the above described property containing, by cal- 
culation, — Acres, — Eoods, and — Perches, more or less. 

Erasmus Kuggles, 
Variation of the needle — ° — ' — " W (or E). Surveyor. 



BOOK IV. 

TRANSIT SURVEYING. 



SECTION I. 

SU RVEYOR'S TRAN SIT. 

177. The transit is an instrument used for measuring hori- 
zontal and (when furnished with a vertical circle) vertical angles. 
It is placed on a tripod, TTT, to which it is screwed fast by 
means of a horizontal brass plate, DE, Upon the upper surface 
of the plate DE rest four screws with milled heads, called level- 
ing screws, which work through the second horizontal plate, FG, 
into cylindrical nuts, shown in the figure. The two plates DE 
and i^(r are called leveling plates. The lower leveling plate, DE, 
is made in two pieces — the upper piece, which is screwed fast to 
the top of the tripod, having a large opening in its centre, in 
which the smaller lower one is shifted from side to side, or 
turned completely around. 

By this arrangement, termed a *' shifting centre," the instru- 
ment is easily moved over the upper plate, and the plummet, 
which hangs from the centre, set precisely over a point without 
moving the tripod. 

The upper side of the plate DE terminates in a curved 
surface, which encloses a ball that is nearly a hemisphere with 
the plane of its base horizontal. This ball or hemispherical nut 
is scrjag5d fast to the smaller base of a solid conic spindle, that 
pas^ tlirough the curved part of the plate DE. To this 
sg||dle is firmly attached the second horizontal plate, FG. 



154 



ELEMENTS OF SURYEYIITG. 



[book IV. 



To this spindle, also, and above the plate FG, is fitted and 
fastened by spring-catch, a socket, called the main socket, to 
which is attached a third horizontal and circular plate, BB. 




Fig. 85. 



called the limb of the instniment. Fitted to the upper 
conical surface of the main socket is a second socket to which 
is united a thin circular plate, A A, called the Vernier Plate, 



SEC. I.J 



surveyor's transit. 



155 



which rests on the limb of the instrument and carries a compass- 
circle, standards, &c., as shown in Fig. 85. 




Fig. 86. 



178. The sectional view, Fig. 86, shows the interior construc- 
tion of the sockets of the transit, the manner in which it is 
detached from the spindle, and the means by which it can be 
taken apart if desired. 

In the figure, the limb BB is attached to the main socket of 
LL, which is itself carefully fitted to the conical spindle P, and 
held in place by the spring catch C. 

The upper plate, A A, carrying the compass-circle, standards, 
&c., is fastened to the flanges of the socket, Z7Z7, which is. fitted 
to the upper conical surface of the main socket L ; the weight of 
all the parts being supported on the small bearings of the end of 
the socket, as shown, so as to turn with the least possible 
friction. 

A small conical centre, in which from below is inserted the 
strong screw, 8, is brought down firmly upon the upper end of 
the main socket LL, and thus holds the two plates of the instru- 



156 ELEMEifTS OF SURVEYING. [BOOK IT. 

ment securely together, while at the same time allowing them to 
move freely around each other in use. 

A small disc above the conical centre contains the steel 
centre-pin upon which rests the needle, as shown ; the disc is 
fastened to the upper plate by two small screws, as represented. 

The main socket with all its parts is of the best bell-metal 
and is most carefully and thoroughly made, the long bearing of 
the sockets ensuring their firm and easy movement, while at the 
same time they are entirely out of the reach of dust, or other 
source of wear. 

When desired the whole upper part of the instrument can be 
taken off from the spindle by pulling out the head of the spring- 
catch at Cy and when replaced will be secured by the self-acting 
spring of the catch. 

The figure also shows the covers of the leveling screws, the 
shifting centre of the lower leveling plate, and the screw and 
loop for the attachment of the plummet. 

179. On the upper surface of the plate FG, Fig. 85, rests a 
clamp which goes round the main socket and which, being com- 
pressed by the clamp-screw K, is made fast to it. This clamp is 
thus connected with the plate FG. Two small cylinders, aa, are 
fastened to the plate FG ; through these cyHnders thumb 
screws, //, called slow-motion screws or tangent screws, pass and 
abut against opposite sides of a piece projecting from the clamp- 
ring, thus preventing the clamp from moving in either direction. 
When the clamp is compressed against the main socket by the 
clamp-screw K, the limb of the instrument is prevented from 
revolving. One of the tangent screws /, being then loosened as 
the other is tightened, will slowly and steadily move the clamp- 
ring itself and with it, of course, the limb. 

The limb, BB, has a silvered circle near its outer edge, on 
which the graduation for horizontal angles is made. In the 



SEC. I.] SURVEYOR'S TRANSIT. 157 

instrument described, the circle is divided into degrees and half 
degrees ; the degrees are figured in two rows, viz : from to 360, 
and from to 90 each way. 

The vernier plate, A A, has two openings, one of which is 
shown in the Figure 85, placed diametrically opposite each other, 
in which, attached to the plate A A and moving with it, are 
small silvered arcs called verniers, which serve to read the limb 
about which they revolve. 

The verniers are double, having on each side of the zero 
mark thirty equal divisions corresponding precisely with twenty- 
nine half-degrees of the limb ; they thus read to single minutes, 
and the number passed over is counted in the same direction in 
which the vernier is moved. 

The use of two opposite verniers gives the means of testing 
the correctness of the graduations, the perfection with which 
they are centered and the dependence which can be placed upon 
the accuracy of the angles indicated. 

Two spirit levels, at right angles to each other, are attached 
to the vernier plate by small adjusting screws, one being fixed 
to the standard which supports the telescope, so as not to obstruct 
the light which falls on the vernier opening beneath. 

The vernier plate turns freely around with the socket to 
which it is attached. It is made fast to the limb by the clamp- 
screw Q, Fig. 85, after which the smaller motions are made by 
the tangent-screw R. 

There is a compass on the vernier plate that is concentric 
with it, the uses of which have been explained under the head 
Compass. It also may be made to serve as a check upon the 
measurement of angles with the transit, as from the magnetic 
bearings of two courses may be found the angle between them, as 
explained in Art. 114. 

180. The standards which support the horizontal axes of the 



158 



ELEMENTS OF SURVEYING. 



[book IV. 



B 




^^|^~W^ = 



telescope with its attached level, rest on and are made fast to the 
Ternier plate. The vertical circle MM, Fig. 85, called the 
Vertical Limb, is securely fastened to 
the axis of the telescope; it is plated 
with silver, graduated to half degrees, 
and with its vernier enables the sur- 
veyor to obtain vertical angles to single 
minutes. There is a clamp-and-tangent 
arrangement connected with the tele- 
scope ; it consists of an arm at one eod 
encircliug the telescope axis, and at the 
other connected with the tangent-screw 
0, Fig. 85. The clamp is fastened at 
will to the axis by a clamp-screw, and 
then by turning the tangent-screw the 
telescope is raised or lowered as desired. 

181. The telescope is from ten to 
eleven inches long, firmly secured to an 
axis having its bearings nicely fitted in 
the standards, and thus enabling the 
telescope to be moved in either direction, 
or turned completely around if desired. 

The different parts of the telescope 
are shown in Fig. 87. 

The object-glass is composed of two 
lenses, so as to show objects without 
color or distortion, is placed at the end 
of a slide having two bearings, one at 
the end of the outer tube, the other in 
the ring CC, suspended within the tube 
by four screws, only two of which are 
shown in the cut. ^m. 8t. 




SEC. I.J 



SURVEYOR'S TRAI^SIT. 



159 



The object-glass is' carried out or in by a pinion working 
in a rack attached to the slide, and thus adjusted to objects 
either near or remote as desired. 

The eye-piece is made up of four plane convex lenses, which, 
beginning at the eye end, are called respectively the eye, the 
field, the amplifying, and the object lenses, the whole forming a 
compound microscope having its focus in the plane of the cross- 
wire ring BB. 

The eye-piece is brought to its proper focus usually by 
twisting its milled end, the spiral movement within carrying the 
eye-tube out or in as desired ; sometimes a pinion, like that 
which focuses the object-glass, is employed for the same purpose. 

182. In order that the telescope may be directed to an object 
with precision, two spider's lines, or fine wires, are fixed at right 
angles to each other, and placed within the barrel of the tele- 
scope, and at the focus of the eye-glass. 

The cross-wire diaphragm is a 
small ring of metal suspended in 
the tube of the telescope by four 
capstan-head screws, ff, gg, Fig. 88. 
The ring can thus be moved in 
either direction by working the 
screws with an adjusting-pin. 
Across the flat surface of the ring 
two fine fibres of spider's web, or, 
better, very fine platinum-wire, are 

extended at right angles to each other, their ends being cemented 
into fine lines cut in the metal of the ring. 

The intersection of the wires forms a very minute point 
whicii, when they are adjusted, determines the optical axis of the 
telescope, and enables the surveyor to fix it upon an object with 
the greatest precision. 




160 ELEMENTS OF SURVETIl^G. [BOOK IV. 

The openiugs in the telescope tube are made considerably 
larger than the screws, so that, when these are loosened, the ring 
may be moved a short distance in the direction of the length of 
either wire for purposes of adjustment as hereafter described, and 
may also be slightly turned for the same purpose. 

183. To measure horizontal angles correctly, the limb of 
the instrument must be made truly horizontal. 

To level the instrument, extend the legs and place them so as 
to bring each bubble as nearly as possible to the middle of its 
tube. Turn the vernier plate till one of the levels is parallel to 
one pair of leveling screws, the other level will be parallel to the 
other pair of screws. If the bubble in either level is not at the 
middle, turn the leveling screws to which that level is parallel, in 
contrary directions, thus raising one side of the horizontal plate 
carrying the graduated circle and lowering the other till the 
bubble is brought to the middle of that tube ; with the other 
pair of leveling screws, bring the bubble in the other level to the 
middle of the tube ; when the bubbles in both levels are at the 
middle of their respective tubes, the plane of the levels, and, 
consequently, the graduated circle by which horizontal angles are 
measured, are truly horizontal. Two opposite leveling screws 
may be turned in contrary directions, by holding each screw with 
the thumb and forefinger, and turning so that both thumbs turn 
in, or both out. When the thumbs are turned outward, the left- 
hand side of the circle is raised ; when both are turned inward, 
the right-hand side is raised. The bubble always runs to the end 
of the level which is too high. 

If the leveling screws become jammed, so as to work hard, 
turn one of them only, forward or backward, till they are free 
again ; sometimes they work hard, from setting the other pair 
tight while at a large angle of inclination with the lower plate, 
in which case the other pair must be loosened again. 



I 



SEC. I.] SURVEYOR'S TRAIS'SIT. 161 

ADJUSTMENTS OF THE TRANSIT. 

Before using the instrument, it must be adjusted; that 
is, the parts must be brought to their proper relative positions. 
There are four principal adjustments. 

First Adjustmeii^t. — To malce the axes of the levels, on the 
linib, perpendicular to the axis of the i7istrument. 

184. Turn the horizontal limb until one of the levels is 
parallel to one pair of leveling screws ; the other will be parallel 
to the other pair. Bring the bubble in each tube to the middle 
with its parallel pair of leveling screws ; after which turn the 
horizontal limb half way round to reverse the levels, and if the 
bubbles remain in the middle of the tubes, the levels are properly 
adjusted. But if either bubble recedes from the centre, that 
level must be adjusted by raising the lower or depressing the 
higher end, one half by the leveling screws, and one half by the 
screws which fasten the level to the plate. Again reverse the 
levels, and if the bubble does not remain in the middle of the 
tube, correct the error as before, and repeat the operation till the 
bubble remains in the middle during an entire revolution of the 
plate. 

Each level must be adjusted separately. 

Second Adjustment. — To fix the intersection of the cross- 
ivires in the axis of the telescope, which is called the line of 
collimation. 

185. Having screw^ed the tripod to the instrument, extend 
the legs, and place them firmly. Level carefully, then loosen the 
clamp-screw Q, of the vernier plate, and direct the telescope to a 
small, well-defined, and distant object. Slide the eye-glass of the 
telescope till the cross-wires are distinctly seen ; then with the 



162 ELEMEi^TS OF SURVEYIi?-G. [BOOK IV. 

thumb-screw, which forces out and draws in the object-glass, 
adjust this glass to its proper focus, when the object, as well as 
the cross-wires, will be distinctly seen ; clamp the vernier plate 
and, by the tangent-screw R, bring the intersection of the cross- 
wires exactly upon a well-defined point of the object. 

Now move the eye from side to side across the eye-glass ; if 
there results any movement of the cross-wires away from the 
point sighted, it is evidence that the image of the point does not 
fall exactly upon the plane of the cross-wires, and the object-glass 
must be again focussed, till no such displacement can be detected. 
This displacement is called ** parallax." 

Having done this, sight some well-defined point, or the mid- 
dle of a pin, distant two or three hundred feet, and having 
clamped tbe horizontal motions, revolve the telescope about its 
horizontal axis and sight a pin in the opposite direction from the 
first and at the same distance from the plumb ; now loosen the 
lower clamp and, revolving the instrument about the vertical 
axis, sight the middle of the first pin again, and clamp ; reverse 
the telescope upon the horizontal axis, and if it now cuts the 
second pin, as before, the adjustment is correct; but if the 
intersection of the wires falls on one side of the second pin, 
bring it back over one quarter of the error, by the capstan screws 
on each side of the telescope over the cross- wire ring. As the 
eye-piece inverts the position of the wires, the operation of loos- 
ening one of the screws and tightening the other on the opposite 
side, must be conducted as if to increase the error observed. 
To insure the accuracy of the adjustment, repeat the whole 
operation as above described, until the reversals cut the pins 
exactly. 

186. The reason for the above directions will be apparent 
upon examination of Fig. 89. 




SEC. I.] SURVEYOR'S TRANSIT. 163 

Let AA' represent the 
line of collimation pro- 
longed both ways from the 
instrument at /, and sup- 
pose that the intersection 
falls at B, the cross- wires not being in adjustment; on reversing 
the telescope the intersection will fall at B', as far to the left of 
A' as B was to the right ot A; if now we turn the instrument 
180°, by the horizontal limb, the intersection will fall at C, to 
the left of A, CA being equal to BA, and on reversal the inter- 
section will fall at C, CA' being equal to A'B' ; if the cross- 
wires be moved over G'A', the adjustment will be accomplished. 
But it is not well to risk introducing possible errors of gradua- 
tion of the horizontal limb, or errors of reading, into the opera- 
tion, and therefore instead of turning the instrument through 
180°, the surveyor turns till he sights B the second time, thus 
passing over 180° ±(7^; this causes C to fall at C", C'G" being 
equal to GB, and therefore B'G" must be /owr times G'A', or 
four times the error of adjustment. 

187. The adjustment of the horizontal wire, which is only an 
approximation to perfect adjustment, is accomplished thus : 
Set the zero of the vertical limb at the zero of its vernier, and 
clamp; sight some sharply defined point upon a wall or staff, 
distant about 200 feet ; unclamp and reverse the telescope upon 
its horizontal axis and set the zero of the vernier at 180° on the 
vertical limb ; now unclamp the lower clamp and revolve the 
instrument upon its vertical axis till the chosen point is again 
in the field of view ; if the point and the intersection of the 
cross-wires do not coincide, correct half the error by the screws, 
moving the cross-wire ring up or down, and the remaining half 
error by the leveling screws ; repeat the operations till the error 
is wholly corrected. 



164 ELE3IEXTS OF SURVEYING. [BOOK IT. 

If the instrument should not be furnished with a vertical 
limb, the adjustment of the horizontal hair can only be guessed 
at, by bringing it to the centre of the field of the telescope. 

Thied Adjustment. — To make the axis of the attached level 
of the telescope parallel to the line of coUimation, 

188. First level the instrument carefully, and with the clamp 
and tangent movement to the axis, make the telescope horizontal 
as near as may be with the eye ; then having the line of collima- 
tion previously adjusted, drive a stake at a convenient distance, 
say from one to two hundred feet, and note the height cut by 
the horizontal cross-wire upon a staff set on the top of the stake. 
Fix another stake in the opposite direction and at the same dis- 
tance from the instrument, and, without disturbing the telescope, 
turn the instrument upon its spindle, set the staff upon the 
stake, and drive in the ground until the same height is indicated 
as in the first observation. The top of the two stakes will then 
be in the same horizontal line, however much the telescope may 
be out of level. 

Now remove the instrument from fifty to one hundred feet to 
one side of either of the stakes, and in line with both ; again level 
the instrument, clamp the telescope as nearly horizontal as may 
be, and note the heights indicated upon the staff placed first 
upon the nearer and then upon the more distant stake. 

If both readings do not agree, correct nearly the whole error 
of the more distant reading by the tangent screw, and continue 
correcting and reading until the two readings agree ; now place 
a stake at a point in line T\ith the two already fixed and about 
one hundred feet beyond the more remote, and mark upon it the 
point cut by the cross-wires, and mark also the corresponding 
point upon the more distant of the first two stakes ; a line 
passing through these two marks will be a horizontal or level 



SEC. I.] SURVEYOE'S TRANSIT. 165 

line with reference to the first position of the instrument, but 
not with reference to its present position. 

Set the instrument again at its first position, and by the 
tangent-screw make the readings, aboye or below the marks, the 
same ; the line of collimation will now be horizontal, and the 
bubble of the attached level must be brought, very carefully, to 
the middle of its run by the screws at the end of the level tube. 

Fourth Adjustment. — To make the axis of the vertical limb 
perpendimilar to the axis of the iiistrument. 

189. Bring the intersection of the cross-wires of the tele- 
scope upon a plumb-line, or any well-defined vertical object, 
and move the telescope with the thumb-screw 0; if the inter- 
section of the cross-wires continues on the vertical line, the axis 
is horizontal. 

Or, the adjustment may be effected thus : Direct the inter- 
section of the cross-wires to a well-defined point that is consider- 
ably elevated ; then turn the vertical limb, until the cross- wires 
cut some other well-defined point, upon or near the ground; 
revolve the telescope on its axis, and turn the vernier plate 180° ; 
then, if in elevating and depressing the telescope the line of 
collimation passes through the two points before noted, the axis 
is horizontai If it be found, by either of the above methods, that 
the axis is not horizontal, it must be made so by adjusting the 
standards which support the telescope. One of these standards 
is made adjustable, and by raising the standard at the end towards 
which the deviation from the vertical occurs, or depressing the 
other, the adjustment is made. 

Note. — In making the third adjustment, the curvature of the earth is 
taken into account that the adjustment may be theoretically correct. 



166 ELEMENTS OF SURVEYIifG. [BOOK IV. 



SECTION II. 

MEASUREMENT OF ANGLES. 

190. To Measure with the Transit a Horizontal Angle 
subtended by two objects. — Place the axis of the instrument 
directly over the point at which the angle is to be measured. 
This is effected by means of a plumb, suspended from 
the centre of the plate which forms the upper end of the 
tripod. 

Haying made the limb truly level, place the of the vernier 
at any exact degree (merely to avoid minutes and seconds in the 
first reading), and fasten the clamp screw Q of the vernier plate. 
Then, facing in the direction between the lines which subtend 
the angle to be measured, loosen the lower clamp and sight one 
of the objects very nearly, without wasting time in trying to 
secure perfect bisection by the cross-wires ; tighten the lower 
clamp and make perfect bisection by the lower tangent screw. 

This being done, loosen the clamp-screw Q of the vernier 
plate, and direct the telescope to the other object ; the arc 
passed over by the point of the vernier, is the measure of the 
angle sought ; the difference of the two readings (if the 0° of 
the limb be not passed in turning to the second object), is the 
required angle ; if the 0° be passed over, then add 360° to the 
smaller reading and subtract the greater reading from this sum. 
Alivays le careful to tahe both readings from the same vernier. 1 

Note 1. — In measuring horizontal angles, it does not matter 
whether the telescope has to be elevated or depressed to sight 
either or both of the objects, since the telescope revolves on its 
axis in a vertical plane, and the angles measured are always the 
horizontal projections of the angle. 



SEC. II.] MEASUREMENT OF ANGLES. 167 

Note. 2. — When great accuracy is desired, a repetition, or 
several repetitions of the measure may be made, and a mean of 
the observations taken as the true measure. (See Arts. 254, 
255.) 

In the measurement of vertical angles, it is necessary to 
understand, first: 

191. The method of determining the index error of the 
vertical limb. Having leveled the horizontal limb, direct the 
telescope to some distinctly marked object, as the top of a 
chimney, and read the instrument. Eevolve the telescope on its 
axis and turn the vernier plate 180°, and having directed 
the telescope to the same object again, read the instrument. 
If the two readings are the same, the limb is adjusted ; that is, 
the of the limb coincides with the of its vernier, when the 
axis of the telescope is parallel to the horizontal limb. 

When the reading, found with the telescope in the first 
position, is greater than that obtained in the reversed position, 
the true elevation of the object, which is equal to a mean 
of the readings, may be obtained by subtracting half the 
difference from the first reading. If the first reading is less 
than the second, the half difference must be added to the 
first. Hence, 

To find the index error, take the reading of the limb when 
the telescope is directed to a fixed object, a)id then with the telescope 
and vernier plate both reversed. Take half the difference of these 
readings, and affect it with a minus sign if the first is the 
greater, or a plus sign if the second is the greater ; this is equal 
to the index error. 

Let the operation be repeated several times, using different 
objects, and a mean of the errors will be more correct than the 
result of a single observation. Then, second : 

192. Having determined the index error, let the axis of the 



168 



ELEMENTS OF SURVEYIi?"G. 



[book IY. 



telescope be directed to any poiut either above or below the 
plane of the limb, and read the arc indicated by the of the 
vernier. To the arc so read apply the proper correction, if any, 
and the result will be the true angle of elevation or depression. 

The angle of elevation may be more correctly found by 
taking the elevation of the object, and repeating the observation 
with the telescope and vernier plate reversed, and then taking 
a mean of the readings for the angle required. 

193. The true azimuth of a line or course is the angle which 
the vertical plane through it makes with the plane of the 
meridian. 

The azimuth of a line or course referred to some preceding 
course, or to any given line, is the angle made by the line or course 
with the prolongation of the line of reference or of a parallel to it 
through the angular point, the measurement being made around 
to the right. Thus, 

The azimuth of BC with AB, 
(Fig. 90), is the angle RBC ; the 
azimuth of CD with ^^ is the angle 
SOD (SQ being parallel to RA). 



^P 




Fig. 90. 



194. To find with the transit 
the azimuths of several successive 
courses with a given first course. — 

Place the transit at B (Fig. 91) and level it ; make the zero of the 
vernier coincide with the zero of the horizontal limb, and clamp 
the vernier plate ; direct the 
telescope to A, and clamp 
the limb; revolve the tele- 
scope on its horizontal axis, 
and it will then point in 
the direction of BE, the 
prolongation of AB ; un- 



R 




SEC. II.] MEASUREMENT OF ANGLES. 169 

clamp the vernier plate and direct the telescope to C ; the 
reading will be the angle RBC, the azimuth of BC with AB. 

Clamp the vernier plate and remove the instrument to C ; 
reverse the telescope on its horizontal axis, loosen the lower 
clamp and sight B ; the horizontal limb now has its zero point 
in the direction of QPy or its parallel AR, as it had at B ; 
tighten the lower clamp and revolve the telescope on its axis ; 
unclamp the vernier plate, direct the telescope to D, and the 
reading will be the angle QCD, which CD makes with PQy or its 
parallel AR, and is the azimuth of CD with AB, 

Clamp the vernier plate and remove the transit to D ; reverse 
the telescope on its horizontal axis, loosen the lower clamp and 
sight C ; the limb will then have its zero point in the direction 
T8, or its parallel AR, as it had at C and B ; tighten the lower 
clamp and revolve the telescope on its axis ; unclamp the vernier 
plate and direct the telescope to E ; the reading will be the 
angle TDE, which DE makes with T8, or its parallel RA, and is 
the azimuth of DE with AB. 

Proceed in like manner with any number of successive 
courses. 

If the courses enclose a field, the reading at the last station, 
sighting to the first station occupied by the transit, should be 360°. 

The course AB, with respect to which the azimuths are 
taken, is called the Meridian of the Survey. 

195. The magnetic bearing of a line or course, is the angle 
which it makes with the magnetic meridian, and its true bearing 
is the angle which it makes with the true meridian. 

In finding the area of a piece of ground, it is not necessary to 
have either the true or the magnetic bearing. It is sufficient to 
have the bearings of the several successive courses with respect to 
one of the courses taken as a meridian. These may be found 
from the azimuths as f oUows : 



170 



ELEMENTS OF SURVEYING. 



[book IV. 




First suppose a north and south line, and 
an east and west line to be drawn, and the 
graduation to be made from 0° to 360°, 
as represented in Fig. 92 ; let the course 
taken as the meridian be represented by the 
line JVS; then when the azimuth of the 
second course with the first, or meridian, is less than 90°, it is 
the bearing, and since the course lies between N and B, the 
bearing is JVU ; when the azimuth is 90°, the bearing is due 
east ; when the azimuth is between 90° and 180°, the course 
lies between S and F, the bearing with the first course, or 
meridian, will be SF, and may be obtained by subtracting the 
azimuth from 180° ; when the azimuth is 180°, the bearing is due 
south ; when the azimuth is between 180° and 270°, the course 
lies between S and W, the bearing is, therefore, SW, and may 
be obtained by subtracting 180° from the azimuth ; when 
the azimuth is 270°, the bearing is due west ; when the 
azimuth is between 270° and 360°, the 
course lies between iV and TF, the bear- 
ing is JVW, and may be obtained by 
subtracting the azimuth from 360°. 

For example: Let AB (Fig. 93) be 
taken as the meridian, and let the 
azimuths of the several courses with 
AB be as in the table ; then will the 
bearings of the several courses with 
AB, be as noted in the 






TABLE. 




station. 


Azimuth with AJB. 


Bearing with AB. 


A 


0° 


North 


B 


93° 30' 


S 86° 30' E 


C 


175° 30' 


S 4°30'E 


D 


257° 


S77° W 






SEC. II.l 



MEASUREMENT OF ANGLES. 



in 



196. If it is desired to find the true or magnetic bearing of a 
course from its azimuth with a given course taken as a meridian, 
it may be obtained by finding the true or magnetic bearing of 
the course taken as meridian and subtracting it from, or adding it 
to, the azimuth of the given course, according as the bearing of 
the reference line "is NW or NE ; the result if less than 90° is 
the true, or magnetic bearing, and is NE ; if more than 90° and 
less than 180°, subtract it from 180°, and the result will be the 
bearing, 8E ; &c. 

For illustration, take the example in the last article, and let 
the magnetic bearing of the meridian course, or reference line, 
AB, be N 31| W. 



TABLE, 



Station. 


Azimuth with AB. 


Bearing with AB. 


Magnetic Bearing. 


A 


0° 


North. 


N n\ w 


B 


93° 30' 


S 86° 30' E 


]S[62° E 


C 


175° 30' 


S 4° 30' E 


S 36° E 


D 


257° 


S 77° W 


S 45° 30' W 



If BCy of which the magnetic bearing is N 62° E, had been 
taken as the reference line, its bearing would have been added to 
the azimuths to obtain tha magnetic bearings of the successive 
courses, as follows : 



Station. 


Azimuth with B C. 


Bearing with BC, 


Magnetic Bearing. 


A 


226° 30' 


S 86° 30' W 


N 31i W 


B 


0° 


North. 


N 62° E 


C 


82° 


N82° E 


S 36° E 


D 


163° 30' 


S 16° 30' E 


S 45i W 



Note. — For 8E and SW in bearing of reference line, read 
iVPTand NE, respectively, in applying the above rule. 



172 ELEMENTS OF SURVEYING. [BOOK IV. 

To find the True Meridian with the Transit. 

197. Before making the observations it will be necessary to 
devise some means by which the cross- wires may be lighted, that 
they may be distinctly visible. 

To do this, take a board of about one foot square, paste 
white paper upon it and perforate it through the centre ; the 
diameter of the hole being somewhat larger than the diameter of 
the telescope of the transit. Let this board be so fixed to a 
vertical staff as to slide up and down freely; and let a small 
piece of board, about three inches square, be nailed to the 
lower edge of it for the purpose of holding a candle. 

About twenty-five minutes before the time of the greatest 
eastern or western elongation of the pole-star, as shown by the 
table of elongations (see Art. 164), let the transit be placed at 
a convenient point and leveled. Let the board be placed about 
one foot in front of the transit, a lamp or candle placed on the 
shelf at its lower edge, and let the board be slipped up or 
down until the pole-star can be seen through the hole. The 
light reflected from the paper will show the cross- wires in the 
telescope of the transit. 

Then let the vertical cross- wire be brought exactly upon 
the pole-star, and, if it is an eastern elongation that is to be 
observed and the star has not yet reached the most east- 
erly point, it will move from the line toward the east, and the 
reverse when the elongation is west. 

At the time the star attains its greatest elongation, it will 
appear to coincide with the vertical cross-wire for some 
time, and then leave it in the direction contrary to its former 
motion. 

As the star moves toward the point of greatest elongation, 
the telescope must be continually directed to it by means of 
the tangent-screw of the vernier plate ; and when the star has 



SEC. II.] MEASUREMEi^T OE AI^GLES. 173 

attained its greatest elongation, great care should be taken that 
the instrument be not afterward moved. 

Next turn the telescope very carefully upon its horizontal 
axis and fix a peg in the ground, distant 150 or 200 feet ; to do 
this, let the light of the lantern shine through a small hole in a 
board, across the centre of which hole a plumb-line hangs, and 
by sighting the line thus seen the peg may be fixed. Also mark 
the point directly under the transit plumb ; the line passing 
through this point and the staff, makes an angle with the true 
meridian equal to the azimuth of the pole-star. 

From the table of azimuths (see Art. 164), take the azimuth 
corresponding to the year and nearest latitude. If the observed 
elongation was east, the true meridian lies on the west of the line 
which has been found, and makes with such line an angle 
equal to the azimuth. If the elongation was west, the true 
meridian lies on the east of the line found; and, in either 
case, laying off the azimuth angle with the transit, gives the 
true meridian. 

198. Another method depends upon the fact that at the 
same angular distances east and west of the meridian the alti- 
tudes of any selected star are equal. Direct the transit to any 
bright star towards the south, and east of the meridian, and 
clamp the vertical limb ; carefully read the horizontal limb ; 
do not disturb the clamp and tangent of the vertical limb in 
any way, but loosen the clamp of the vernier plate, and after a 
suflBcient interval of time, two or three hours perhaps, as it is 
not good practice to set the altitude when the star is too near 
the meridian, turn the plate upon its spindle towards the west ; 
bring the star upon the cross-wire intersection as its altitude 
decreases, moving the vernier plate only ; it will now have the 
same altitude as before; read the horizontal limb ; take half the 
difference of the two readings, and set this half angle back 
towards the east ; the line thus determined is the meridian. 



174 



ELEilEXTS OF SURVETINTG. 



[book IV. 




Fig. 94. 



APPLICATIONS TO HEIGHTS AND DISTANCES. 

199. To determine the horizontal distance to a point 
which is inaccessible by reason of an intervening river. — 
Let C be the point (Fig. 94). Measure 
along the bank of the river a hori- 
zontal base-line AB, and select the 
stations A and B, in such a manner 
that each can be seen from the other, 
and the point C from both of them. 
Then measure the horizontal angles 
CAB and CBA, with the transit. 

Let us suppose that we have measured 

AB =zmO yards ; 

CAB = A=bT 35", 

and CBA = B = 6^° 51'. 

Then, C= 180° - {A + B) = 57° 34'. 

To find the distance BC. 

sin C : sin A :: AB : BC. 

Applying logarithms, we have, 

(a. c.) log sin C (57° 34') .... 0.073649 

log sin ^ (57° 35') .... 9.926431 

logAB (600) 2.778151 

log ^6' 600.11 2.778231 



To find the distance AC. 
sin C : sm B :: AB : AC, 
and applying logarithms, we have, 



SEC. II.] MEASUKEMEKT OF ANGLES. 

(a. c.) log sin (7 (57° 34') .... 0.073649 
log sin ^ (64° 51') .... 9.956744 
logAB (600) 2.778151 

log^C 643.94 2.808544 



175 



B< 




Fig. 95. 



To determine the altitude of an inaccessible object above a 

given horizontal plane. 

FIRST METHOD. 

200. Suppose D to be an inac- 
cessible object, and BC the horizon- 
tal plane from which the altitude is 
to be measured ; then, if we suppose 
DC to be a vertical line, it will repre- 
sent the required distance. 

Measure any horizontal base-line, 
as BA ; and at the extremities B and A, measure the horizontal 
angles CBA and CAB. Measure, also, the angle of elevation 
DBC, ' 

Then, iu the triangle CBA, there will be known two angles 
and the side AB ; the side BC can therefore be found by 
calculation. Having found BC, we shall have, in the right- 
angled triangle J)BC, the base BC and the angle at the base, 
to find the perpendicular DC, which measures the altitude of 
the point D above the horizontal plane BC. 

Let us suppose that we have found, by measurement, 

BA = 780 yards. 

The horizontal angle CBA = B = 4.1° 24', 
the horizontal angle CAB = A = m° 28', 
and the angle of elevation VBC^ 10° 43'. 



l'^6 ELEMENTS OF SUEVEYING. [BOOK IT, 

To find, in the triangle BCA, the horizontal distance BC. 
The angle BCA = C = 180° — {A + B) = 42° 08'. 
Then, sin (7 : sin .4 :: AB \ BC', 

and applying logarithms, we have, 

(a. c.) log sin C (42° 08') .... 0.173369 

log sin A (96° 28') .... 9.997228 

log^^ (780) 2.8920 95 

\ogBC 1155.29 yards . . . 3.062692 

In the right-angled triangle DCB, to find DC. 
We have, from Theorem IV, 

R : tan BBC :: BC : DC. 

Applying logarithms, we have, 

(a. c.) logi? (90°) .... 0.000000 

log tan i)^C (10° 43') . . . 9.277043 
logBC (1155.29) . . . 3.0626 92 

log DC 218.64 .... 2 .339735 

Note 1. — It might, at first, appear, that the solution given 
requires that the points B and A should be in the same 
horizontal plane ; but it is entirely independent of such a 
supposition. 

For, the horizontal distance represented by BA is the same, 
whether the station A is on the same level with B, above it, 
or below it. The horizontal angles CAB and CBA are also 
the same, so long as the point C is in the vertical line DC. 
Therefore, if the horizontal line through A should cut the ver- 
tical line DC, at any point, as B, above or below C, AB would 
still be the horizontal distance between B and A, and AB 
would be the horizontal distance between A and C. 



If 



SEC. II.] [EASUKEMENT OF ANGLES. 177 

If at Ay we measure the angle of elevation at the point D, 
we shall know in the right-angled triangle DAE, the base AU 
and the angle at the base ; from which the perpendicular DB 
can be determined. 

Let us suppose that we had measured the angle of elevation 
DAEy and found it equal to 20° 15'. 

First: In the triangle BAC, to find AC, or its equal AE. 
sin (7 : sm B :: AB : AC or AB. 

Applying logarithms, we have, 

(a. c.) log sin C (42° 08') .... 0.173369 
log sin B (41° 24') .... 9.820406 
log AB (780) 2.8920 95 

log AB 768.9 ...... 2.885870 



In the right-angled triangle DAE, to find DE. 
We have, from Theorem IV, 



E : t3inA : : AB 

(a. c.) logi^ (90°) . . 
log tan A (20° 15") . 
log^^ (768.9) . . 

logZ>^ 283.66 . . 



Now, since DC is less than DB, 
it follows that the station B is above 
the station A. That is, 
DB— DC = 283.66 — 218.64 = 
65.02 = BC, 
which expresses the vertical distance 
that the station B is above the station A, 



DB : hence, 



0.000000 
9.566932 
2.8 85870 

2.452802 




178 



ELEMEIfTS OF SURVEYmG. 



[book IV. 



Note 2. — It should be remembered that the vertical distance, 
which is obtained by the calculation, is estimated from a hori- 
zontal line passing through the eye, at the time of observation. 
Hence, the height of the instrument is to be added, in order to 
obtain the true result. 



SECOND METHOD. 




201. When the nature of the ground will admit of it, 
measure a base-line AB, in the direction of the object D, Then 
measure, with the instrument, the angles of elevation at A 
and B. 

Then, since the out- 
ward angle BBC is equal 
to the sum of the angles 
A and ADB, it follows 
that the angle ABB is 
equal to the difference of 

the angles of elevation at A and B. Hence, we can find all the 
parts of the triangle ABB. Having found BB, and knowing 
the angle BBC, we can find the altitude BC. 

This method supposes that the stations A and B are on the 
same horizontal plane; and therefore it can only be used when 
the line ^^ is nearly horizontal. 

Let ns suppose that we have measured the base-line and the 
two angles of elevation, and found, 

AB = 975 yards, A = 15" 36', and DBC= 27° 29'; 

required the altitude DC, 

Arts, i)a= 587.61 yards. 




SEC. II.] MEASUREMENT OF AN"GLES. 179 

To determine the perpendicular distance of an object below 
a given horizontal plane. 

202. Suppose G to be directly 
over the given object, and A the 
point through which the horizontal 
plane is supposed to pass. 

Measure a horizontal base-line 
AB, and at the stations A and B 
conceive the two horizontal lines 
AG, BG, to be drawn. The oblique fig. 98. 

lines from A and B, to the object, are the hypothenuses of two 
right-angled triangles, of which AG, BG, are the bases. The 
perpendiculars of these triangles are the distances from the 
horizontal lines AG, BG, to the object. If we turn the triangles 
about their bases AG, BG, until they become horizontal, the 
object, in the first case, will fall at G', and in the second at G". 

Measure the horizontal angles GAB, GBA, and also the 
angles of depression G'AG, G"BG. 

Suppose that we have measured, and found AB =: 672 yards ; 
BAG= 72° 29'; ^^(7= 39° 20'; angle of depression G'AG = 
27° 49', and G"BG=ir 10'. 

First: In the triangle ABG, the horizontal angle AGB =: 
lSO°-{A-JrB) = 180°-111° 49' = 68° 11'. 

To find the horizontal distance AC. 
sin G : sin B :: AB : AG; hence, 

(a. c.) log sin G (68° 11') 0.032275 

log sin B (39° 20') 9.801973 

log^^ (672) 2.8273 69 

\ogAG 458.79 2.661617 



180 ELEMEJ^TS OF SURVEYING. [BOOK lY. 

To find the horizontal distance BC. 

sin C \ sin .1 :: AB : BC ; whence, 

(a. c.) log sin C (68° 11') 0.032275 

log sin A (72° 29') 9.979380 

log^^ (672) 2.8273 69 

log^C 690.28 2.839024 

In the right-angled triangle CAC, to find CC 
We have, Theorem IV., 

R : idJiA :: AC : CC; whence, 

(a. c.) log R (90°) 0.000000 

log tan A (27° 49') 9.722315 

log AC 458.79 2.661617 

log CC 242.06 2.383932 

In the triangle CBC", to find CC". 

We have, Theorem IV., 

R : tsinB :: BC : CC" j whence, 

(a. c) logR (90°) ...... 0.000000 

log tan B (19° 10') 9.541061 

log^C (690.28) 2.839024 

log CC" 239.93 2.380085 

Hence, also, CC — CC = 242.06 — 239.93 = 2.13 yards; 
which is the height of station A above station B, 



i 



i 



SEC. II.J 



MEASUREMENT OF ANGLES. 



181 



203. P R O B L E M S . 

1. Wanting to know the distance between two inaccessible 
objects, which he in a direct level line from the bottom of a 
tower 120 feet in height, the angles of depression are measured 
from the top of the tower, and are found to be, of the nearer 
57°, and of the more remote 25° 30' ; required the distance 
between the objects. Ans. 173.656 feet. 



2. In order to find the distance between 
two trees, A and B, which could not be 
directly measured because of a pool which 
occupied the intermediate space, the dis- 
tances of a third point C from each of them 
were measured, and also the included angle 
A CB J it was found that. 




CB 


= 672 


yards, 


CA 


= 588 


yards. 


ACB 


= 55° 


40'; 



required the distance AB. 



Ans. 592.967 yards. 



3. Being on a horizontal plane, and wanting to ascertain the 
height of a tower standing on the top of an inaccessible hill, 
there were measured, the angle of elevation of the top of the 
hill 40°, and of the top of the tower 51° ; then measuring in a 
direct line 180 feet farther from the hill, the angle of elevation 
of the top of the tower was 33° 45' : required the height of the 
tower. 

Ans. 83.998. 

4. Wanting to know the horizontal distance between two 
inaccessible objects JE and W, the following measurements were 
made; 



182 



ELEMENTS OF SURVEYIIS'G. 



[book IV. 



^ 



Ei\ 





' AB = 536 yards 
BAW= 40° 16' 
viz.: \ WAE = 57° 40' 
ABE = 42° 22' 
EBW = 71° 07'; 

required the distance EW, 

Ans. 939.617 yards. fig. loo. 

5. Wanting to know the 
horizontal distance between two 
inaccessible objects A and B, 
and not finding any station from 
which both of them could be 
seen, two points C and D were 
chosen at a distance from each 
other equal to 200 yards ; from the former of these points A 
could be seen, and from the latter B, and at each of the points 
C and D a staff was set up. From C a distance CF was measured, 
not in the direction DC, equal to 200 yards, and from D a 
distance DE equal to 200 yards, and the following angles taken, 

^ AFC = 83° 00', BI)E = 54° 30', 
viz.: \ ACD = 53° 30', BDC = 156° 25', 
^AGF = 54° 31', BED = 88° 30'. 

A?is. AB = 345.459 yards. 

6. From a station P there can be seen three objects. A, B, 
and (7, whose distances from each other are known : viz., 

AB = 800, AC= 600, and BC= 400 yards. 

Now, there are measured the horizontal angles, 

AFC = 33° 45', and BFC = 22° 30' ; 

it is required to find the three distances, PA, PC, and PB» 



SEC. II.] 



MEASUREMEN^T OF ANGLES. 



183 



GEOMETRICALLY. 



i 



With the three given sides construct 
the triangle ABC. Then, at A lay off 
the angle BAD z=z 22° 30', and at B the 
angle ABD = 33° 45', and note D, the 
point at which the two lines intersect. 

Through the points A, D, and B, de- 
scribe the circumference of a circle, and 
through G and D draw the line GDP \ 
the point P in which it intersects the 
circumference, will be the position of the station. 

By observing the equal angles in the figure, the trigono- 
metrical solution is not difficult. We find, 




Fig. 102. 



(PA = 710.198 yards. 
Ans. \PG — 1042.524 '' 
PB= 934.289 " 



Note. — This problem is much used in maritime surveying, 
for the purpose of locating buoys and sounding-boats. The 
trigonometrical solution is somewhat tedious, but the geomet- 
rical solution is very easy, as shown above. 



184 ELEMENTS OF SURVEYING. [BOOK IV. 

SECTION III. 

RANGING OUT LINES, ETC. 

204. To range out a line with the transit, place the instru- 
ment, carefully adjusted, over the first station ; direct the 
telescope to a distant and well-defined point in the desired line, 
and clamp both the vernier plate and the horizontal limb. 
The line of sight of the telescope is then in the vertical plane 
of the given line, so that points on the surface bisected by the 
intersection of the cross- wires will be in the required line; let 
an assistant, directed into the line by the observer at the transit, 
fix ranging-rods, or stakes conspicuously marked, as far as the 
power of the telescope extends. Eemove the transit to the 
third or fourth stake from the last set, and place it precisely 
over that position by plumb-bob, and adjust it for observation ; 
the telescope is ranged in the line by sighting backwards and 
forwards to the stakes already set. The line is then continued 
as before. 

If great accuracy be required, each operation must be 
repeated with telescope reversed, as only in this way can error 
in adjustment of cross-wires be eliminated. If the sighting 
with reversed telescope does not agree with the former sighting, 
ake a point midway between the two points sighted, as a point 
in the required line. 

205. A line may be traced in forests or plantings, in which 
there are no general surface obstructions, by the aid of auxiliary 
parallel lines. 

In the illustration, Fig. 103, aa, Ih, cc, are the parallel lines, 
and aa, cc, the auxiliary lines. AB is a line in which h is a given 
point. The distances ab, he, in this line should be measured. 



SEC. III.] 



BANGII^G OUT LINES. 



185 



and also the angle which hh makes with AB. The line hb and 
the auxiliary lines should be traced on this angle, until the 
trace of one of the parallel lines be obstructed by a tree. 



Bs ^, B3 







x.<^~- 







XJ. 



Fig. 103. 






such as hi at (1). Immediately on passing the obstruction 
on one of the auxiliary lines, a line should be traced on the 
measured angle or its supplement, as may be required, and 
traced to intersect the other auxiliary line. The angle made by 
these lines should be measured at the point of intersection to 
verify the trace of the intersecting lines. From these angular 
points in the auxiliary lines, distances to the point (1), equal to 
ab and cb, respectively, should be measured in the transverse 
line, and found to meet, but not overlie, one another. Then 
will the point of meeting in the transverse line be a forward 
point in the line bb. At a suitable distance forward from which 
the point (1) may be observed, a like determination of another 
point in bb should be made. The trace of the line bb should be 
taken upon these points and continued in connection with the 
auxiliary lines until the trace of one of the lines be obstructed, 
such as the line aa at (2). The trace of the obstructed line 
Bhould be taken up by measurements in the transverse line 



186 



ELEMENTS OE SURVEYING. 



[book IV. 



B^Ai, and in a forward parallel line, and the traces continued as 
described above. In like manner the obstructions at (3), (4), 
&c., may be passed, and the trace of the line continued for 
considerable distances with sufficient accuracy for most practical 
purposes. The continued parallelism of the lines at the meas- 
ured distance apart will be a sufficient verification (Smith's 
Treatise on Land Surveying.) 

206. To measure distances by means of the transit. — The 

cross- wire ring in the telescope of the transit is often fitted with 
an arrangement called "Stadia," or *^ Micrometer." 

The Stadia, or Mi- 
crometer, is a compound 
cross- wire ring or dia- 
phragm, shown in Fig- 
ures 104 and 105, having 
three horizontal wires, of 
which the middle one is 
cemented to the ring as 
usual, while the others, 
bb and cc, are fastened to 

small slides, held apart by a slender brass spring hoop, and 
actuated by independent screws, dd, by which the distance 
between the two movable wires can be adjusted to include a 
given space upon a rod, held perpendicular to the optical axis 
in front of the object-glass, at a distance from it equal to its 
principal focal distance. 

When the wires are thus adjusted to include a certain space, 
as two feet for instance, upon a rod placed at a distance of 
100 feet from the specified point on the optical axis, it is found 
that they will cover one foot at half that distance, or four feet 
at a distance of 200 feet ; thus the distance is proportional to 
the space intercepted upon the rod. By adding to the distance 




Fig. 105. 



SEC. III.] BANGING OUT LINES. 187 

thus obtained, the principal focal distance of the object-glass, 
plus the distance of the object-glass from the middle of the 
horizontal axis, the distance of the rod from the station can be 
ascertained without the use of a chain. 

The focal distance of the object-glass can be readily obtained 
by sighting some very distant object, being careful to correct 
instrumental parallax (Art. 185), and then measuring the dis- 
tance from the object-glass to the capstan screws of the cross- 
wire ring, which call a ; now sight some object, distant about 
100 feet, and measure the distance from the object-glass to the 
horizontal axis, which call h ; the sum of these, a-{-bf will be a 
constant, sufficiently exact, to be added to all distances obtained 
by readings on the rod. 

The spaces upon the rod used should be equal to that which 
the instrument intercepts at 100 feet from the point in front of 
the object-glass, and should be numbered from the bottom up ; 
each space should be subdivided into hundredths. The rod 
should have two movable targets, like those used upon leveling 
rods, and should also be furnished with an attached plumb, or 
level, to insure its vertical position. A distinct mark should be 
made upon it at the ordinary height of the horizontal axis of 
the instrument. 

In using the micrometer, sight the middle horizontal hair to 
the ^* height of instrument " mark^ and then direct the targets to 
be moved successively till they coincide with the micrometer 
wires, the rod being kept vertical. If the telescope has been 
level during this operation, the distance given by the rod plus 
the instrument constant, can be recorded ; but if the line of 
sight has been elevated or depressed, then from the distance 
given by the rod, including the instrument constant, must be 
subtracted the product of this distance and the square of the 
sine of the angle of deviation from the horizontal. 

If distances greater than 600 feet are to be measured, the 



188 ELEMENTS OF SURYEYING. fBOOK IV. 

unit of the rod must be less than ^V of the standard distance, 
to avoid the use of a rod too long for practical management. 

If the distances are to be recorded in chains and links, set 
the stadia staff at 66 feet in order to obtain its unit, and then 
graduate it to this unit, and subdivide to hundredths. 

The stadia staff should be spiked, that it may be thrust into 
the ground to secure steadiness. 

The telescope should be a good one, giving a very sharp, 
clear definition of objects, and the micrometer wires should be very 
fine indeed in order to secure close readings. 

The degree of accuracy that may be attained is shown by the 
following table, deduced from that given in ^^Cours de Topo- 
graphic, par A. Lehagre, 1881," to which the student is referred 
for a very full description of the method : 

Focal length of object-glass in 

inches 6 8 10 12 14 

Distance in feet which may be 

safely measured .... 600 800 1000 1200 1400 
Eelative error in horizontal dis- 

tnnpp 1 1 1 1 1 

UdilOC -g-^ -g-g-Q Yo^oO 1800 1400 

With a telescope of ten inches focal length or over, and 
within the above limits, the stadia measurements are as reliable 
as chain measurements on fairly level ground, and are much 
more accurate than chain measures on bad, broken country. 
As the use of the micrometer requires care and consumes time, 
it is not recommended for short distances, except on bad ground, 
swamps, inaccessible distances, &c. 

207. In connection with the chain or tape, the transit is used 
to obtain horizontal distances on sloping ground. The chaining 
is made on the surface of the sloping ground, and not by 
elevating the chain as described in Art. 71 ; and the angle of 



SEC. III.] RANGING OUT LINES. 189 

the slope is taken with the transit, by marking on a rod the 
vertical distance from the horizontal axis of the telescope to 
the ground, and sighting to this mark on the rod held vertical 
at the end of the line measured ; the horizontal distance is equal 
to the measured distance multiplied by the cosine of the angle 
of the slope. 

208. To survey a line, such as a road, boundary of an estate, 
&c., measure the angle of deviation which each line makes with 
the preceding line prolonged, or measure the azimuths which 
each line makes with the first line taken as a meridian (Art. 194), 
and measure, also, the length of each line and offsets to prominent 
objects. Care must be taken to centre the instrument exactly 
over each angular point, as any error in centreing will cause an 
error in the apparent direction of the object sighted, which will 
be the greater the nearer the object is to the instrument. 

209. To survey the streets of a town or city, place the transit 
at the intersection of two or more of the principal streets, 
through which the longest lines of sigh t can be had ; find the 
angle which each of the streets diverging from this point makes 
with the principal street, and find, also, the angle of slope of 
each of the streets at this point ; measure, with the chain or 
stadia-rod, or both as checks one upon the other, the lengths 
of the lines of sight, and take offsets to the corners of all streets, 
to public buildings and prominent objects ; remove the transit 
to the next street and take the angles, angle of slope, measure- 
ments, and offsets as before, and so continue till the survey is 
complete. 



190 



ELEMENTS OF SURVEYING. 



[book IV. 



SECTION IV. 



FARM SURVEYING BY TRANSIT. 

210. The figure and area of any piece of ground raay be 
found by beginning at any one of the angular points and going 
entirely around the boundary, measuring the length of the sides 
by chain or stadia-rod, and the angle which each side makes with 
the preceding side prolonged, called Angle of Deviation, or the 
azimuths of the several sides with a given first side as meridian. 
Let the farm to be surveyed be the one given in Article 123. 
Let the side AB be taken as the meridian of the survey. 
Measure with the transit the azimuths of the several successive 
sides with AB, as directed in Art. 194, and enter them in the 
notes at the left of the station mark. In the following illustration 
both the azimuth and the angle of deviation have been entered, 
though the surveyor would use but one, together with the 
bearings, which should always be entered as a check : 



Azimuths. 


Angles of Deviation, 




Bearings. 


99° 30' 


123° 30' / D 


A 

22.89 


N 30° 35' E 

(N 87° 5' E) 


336° 00' 


100° 55' \ 


A 
1.40 


S 87° 5' W 
(S 7° 50' W) 


76° 55' 


76° 55' / B 


A 
31.95 


N8° E 

(S 68° 55' E) 




A 


A 


N 68° 55' W 



From the azimuths determine the bearings of the several sides 
with AB, as directed in Art. 195, and let them be as noted 
on the following page ; complete the table and determine the area 
as in a compass survey : 



SEC. IV.] 



FARM SURVEYIXG BY TRANSIT. 



191 







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192 



ELEMENTS OF SURVEYING. 



[book IV. 



211. If the surveyor does not record compass-bearings, as 
advised to do in last article, the notes may be kept in precisely 
the same manner as the field notes of the compass survey, as 
shown in Article 123 — substituting azimuths for the hearings 
of the seyeral courses, and angles instead of hearings for the 
prominent objects sighted to. 

212. If the angles of deviation had been measured 
instead of the azimuths, the method of finding the area would 
have been substantially the same. The azimuths would have 
been determined from the angles measured, and the rest of the 
computation would have remained the same. The azimuth of 
any course with a course taken as meridian may be found from 
the angles of deviation, thus : The azimuth of the course taken 
as meridian is zero ; the azimuth of the second course is equal 
to its angle of deviation ; the azimuth of any succeeding course 
is equal to the azimuth of the preceding course increased by the 
angle of deviation of the course itself, as a simple diagram 
will show\ if the interior angles had been measured, the 
angles of deviation would have been obtained by subtracting 
each interior angle from 180° and the method would have been* 



stations. 


Interior Angles. 


Angles of Deviation. 


Azimuth with A B. 


A 


98° 55' 


81° 05' 





B 


103° 05' 


76° 55' 


76° 55' 


C 


280° 55' 


— 100° 55' 


(-24°) 336° 


D 


56° 30' 


123° 30' 


99° 30' 


E 


168° 30' 


11° 30' 


111° 


F 


135° 


45° 


156° 


G 


151° 35' 


28° 25' 


184° 25' 


H 


186° 30' 


-6° 30' 


177° 55' 


I 


136° 


44° 


221° 55' 


K 


123° 


57° 


278° 55' 



The other 
columns as 
before. 



SEC. IV.] FARM SURVEYING BY TRANSIT. 193 

213. When an angle of deviation lies within the boundary of 
the survey, as at C, (see page 87), it must be called, negative ; 
when it lies without the boundary, it is an exterior angle of the 
polygon and is positive. The azimuth obtained, by the rule may 
be negative (as it is in the case of the course CD) ; but as the 
azimuth must be positive, 360^ must be added, to the negative 
result obtained to get the true azimuth. 

The algebraic sum of the angles of deviation should be equal 
to or differ but little from 360°, which fact serves to check the 
correctness of the angles recorded. 

214. Where all the comers of a field may be seen from one 
of them, the area may be determined, as follows : 

Place the transit over the point A, 
Fig. 106, from which the other comers 
may be seen, and measure the angles BA C, 
CAD, and DAK Measure the lengths of 
the diagonals AC and AD, and of the 
sides AB and AB; then the areas of the 
triangles ABC, CAD, and DAB, may 
be separately found from the principle 

that the area of a triangle is equal to half the product of two 
sides and the sine of their included angle ; the sum of the areas 
of the triangles is equal to the area of the field. 

215. If a description of the property is also required, the 
bearing of some one of the lines must be taken, and from this 
and the recorded angles and distances, the "Bearings and 
Courses" may be computed. 

216. Where all the comers of the field may be seen from a 
point within, as A, Fig. 107. Place the transit over the point A ; 
measure the angles BAC, CAD, &c., at A, and the length 




194 



ELEMENTS OF SURVEYING. 



[book IV, 



of the lines AB, AC, &c., from 
A to the several corners of the 
field ; find the areas of the tri- 
angles BAG, CAD, &c., as in 
the last article, and add them 
together. If the boundary is 
irregular, as represented in the 
figure, measure offsets, calculate 
the contents of these smaller 
portions separately, and add or 

subtract them as may be necessary, to find the true area of 
the tract. 




Fig. 107. 



217. — E X A M P L E S . 

1. Eequired the contents and plot of a piece of land, of 
which the following are the field notes : 



stations. 


Azimuths with AB. 


Distances. 


A 


0° 


15.8 ch. 


B 


34° 


17.4 





70° 30' 


32.56 


D 


154° 30' 


14.88 


E 


189° 


24.96 


F 


230° 


14. 


G 


279° 


32.8 


A 


360° 





2. Eequired the contents and plot of a piece of land, of which 
the following are the field notes : 



SEC. IV.] 



FARM SURVEYING BY TRANSIT. 



195 



Stationp. 


Interior Angles. 


Distances, 


A 


92° 30' 


31.80 ch. 


B 


94° 30' 


2.08 


C 


155° 15' 


2.21 


D 


179° 30' 


35.35 


E 


94° 15' 


21.10 


F 


104° 


31.30 



3. Required the area of a piece of land, of which the follow- 
ing measurements were made : 

AB = 20 chains 
AC = 22.57 
AD = 28.64 
AE = 40.80 
AF = 30.95 

angle BAC = 37° 

CAD = 46° 45' 
DAB = 42° 15' 

BAB = 26 Fig. 108. 

Offsets from the line AB were taken as tollows: 




At 3.00 ch. 


E offset 2.50 ch. 


« 6.50 " 




' 0.00 " 


" 9.50 " 


L ' 


' 1.60 " 


"13.00 " 


B 


' 2.00 " 


"16.00 " 


B 


' 1.40 " 


"20.00 " 




* 0.00 « 



BOOK V. 

LAYING OUT AND DIVIDING LAND, 



SECTION I. 

OF DIVIDING LAN D. 

218. The surveyor is often required to lay off a given 
quantity of land, in such a way that its bounding lines shall 
form a particular figure, viz., a square, a rectangle, a triangle, 
&c. He is also often called upon to divide given pieces of 
land into parts containing given areas, or, into areas bearing 
certain relations to each other. 

The manner of making such divisions must always depend 
on a skilful and judicious application of the principles of 
geometry and trigonometry to the particular case. 

For example, if it were required to lay out an acre of 
ground, in a square form, it would be necessary to find, by 
calculation, the side of such a square, and then trace, on the 
ground, a figure bounded by four equal sides at right angles 
to each other. 

219. To lay out a given quantity of land in a square form. 

Rule. — Reduce the given area to square chains or 
square rods ; then extract the square root, and the result 
will he the side of the required square. This square 
being described on the ground, will be the figure required. 



SEC. I.] 



OF DIVIDING LAND. 



197 



220. To lay out a given quantity of land in a rectangular 
form, when one of the sides of the rectangle is given. 

EuLE. — Divide the given area, reduced^ to square chains 
or square rods, by the given side of the required rectangle, 
and the quotient will he the other side. Then, trace the 
rectangle on the ground. 

221. To run a line from the vertex of a triangular field which 

shall divide it into two parts, having to each other the 
ratio of m to 72. 

Let ABC hQ any triangular field. 
Divide the side BC into two parts, 
such that (Geom., Bk. IV, Prob. I) 

BD \ DC \\ m : n; 

and draw the line AD ; 

then will ABD : DAC :: m : n. 

For, the two triangles ABD, ADC, having the same alti- 
tude, are to each other as their bases (Geom., Bk. IV, P. 6, C.) ; 
hence, the triangle is divided into parts having the ratio of 
m to n. 




222. To run a line parallel to one side of a triangular field, 
that shall form with the parts of the two other sides a 



m 



triangle equal to the - part of the field. 



Let CBA represent a triangu- 
lar field, and CA the side parallel 
to which the dividing line is to be 
drawn. 



On the side BC, take 55' equal 




Fig. no. 



198 



ELEMENTS OF SURVEYING. 



[book V. 



/ 7W / fn 

to BC \ — , and on the side BA. take BF equal to BA \ — ; 
11 ^ ^ n 

the line EF is the line required ; for, since it divides the sides 

^Cand BA proportionally, it is parallel to the side CA (Geom., 

Bk. IV, P. XVI) ; and from the similar triangles, we have 

(Geom., Bk. IV, P. XXV), 



or 



BEF : BCA 
BEF ■ BCA 



BE' : BC 



73 



hence, 



777 



7)1 



n ; 



BEF = - BCA. 
n 



Example. — Let it be required to divide the 
triangular field GAB, in which ^C=9ch., 
AB =z \1 q\\., and CB =. 7 ch., into two such 
parts that ADE shall be one-fourth of the 
whole field. 



In this case, we have, 




Fio. 111. 



n 



hence. 



wi = 1, 
AE=4: ch. 50 1. 



4, and ^/^^^/\ = \. 



and AD — 5 ch. 50 1. 



223. To run a line from a given point in the boundary of a 
piece of land, so as to cut off, on either side of the 
line, a given portion of the field. 

Make a complete survey of the field, by the rules already 
given. Let us take, as an example, the field whose area is 
computed in Ex. 1, Art. 140. That field contains 105 ^. 2 R. 
33 P., and Fig. 112 is a plot of it. 

Let it now be required to run a line from station A in 
such a manner as to cut off, on the left, any part of the field ; 
say, 26 A. 2 R. 31 P, 



SEC. I.] 



OF DIVIDING LAND. 



199 




Fig. 112. 



It is seen, by exam- 
ining the field, that the 
division line will proba- 
bly terminate on the 
course CD. Therefore, 
draw a line from A to 
C, which we will call 
the first closing line. 

The bearings and 
lengths of the courses AB, BC, are always known, and in the 
present example are found in the table Art. 140, Ex. 1 ; hence, 
the bearing and distance from G to A can be calculated by Art. 
142 ; they are, in this example, 

Bearing, S 9° 28' E; Course, 23.22 ch. 

Having calculated the bearing and length of the closing 
line, find, by the general method, the area which it cuts off; 
that area, in the present case, is 14 ^. OR. 26 P. 

It is now evident that the division line must fall on the 
right of the closing line AG, and must cut off an area AGH, 
equal to the difference between that already cut off, and the 
given area; that is, an area equal to 12 A. 2 R. b P, 

Since the bearing of the next course CD, and the bearing 
of the closing line A C are both known, the angle A CD which 
they form with each other, can be calculated, and is in this 
example, 79° 32'. Hence, knowing the hypothenuse AG, and 
the angle AGG at the base, the length AO, the perpen- 
dicular let fall on the course CD can be found, and is 22.82 
chains. 

The base of a triangle is equal to its area divided by half 
the altitude. Therefore, if the area 12 y1. 2 i?. 5 P. be re- 
duced to square jhains, and divided by 11.41 chains, which 



200 ELEMENTS OF SURVEYING. [BOOK V. 

is half the perpendicular AG, the quotient, which is 10.95 
chains, will be the base CH. Hence, if we lay off from C, on 
CD, a distance CH, equal to 10.95 chains, and then run the 
line AH, it will cut off, from the land, the required area, viz., 

26 A. 2 B, 31 F, 

Note 1. — If the part cut off by the first closing line 
should exceed the given area, the division line will fall on the 
left of AC. 

Note 2. — If the difference between the given area and the 
first area cut off, divided by half the perpendicular AG, gives 
a quotient larger that the course CD ; then, draw a line from 
A to D, and consider it as the first closing line, and let fall a 
perpendicular on D£J, 

Note 3. — When the point from which the division line is to 
be drawii falls between the extremities of a course, divide the 
course into two parts, at this point. Then consider one of the 
parts as an entire course, and the other as forming a new coarse, 
having the same bearing. The manner of making the calcula- 
tion will then be the same as before. 



224. To cut off from a field, a given area, by a line running 

in a given direction. 

In this case, as in the previous one, a complete and correct 
survey is first necessary. Then, when the whole area is known, 
the position of the line may be approximately determined by the 
inspection of a correct map of the whole. 

We will take, for illustration, Example 4, Art. 140, of which 
Fig. 113 is the plot. 



SEC. I.] 



OF DIVIDING LAND. 



201 



Let it be required to cut off from this area, 50 acres, by a 
line whose bearing shall be S 60° E, or N 60° W. 

We will make a trial of a line starting at 25 chains from 
station 6, on the 6th course. We will call this station A, and 
the trial line AB. 

In order to determine if the area cut off is equal to the 
required area, we must first determine the length of ^^ and 
of B 6. These cannot be determined by the method of sup- 
plying lost notes. 

We must first calculate the length of a line, starting at the 
proposed point, and running to the station nearest to the other 
extremity of the closing line. In this example, from A to 5. 
This is easily found to be 36.406 chains, and its bearing 
N 81° 13' E. 




Fio. 118. 



202 



ELEMENTS OF SURVEYIN^G. 



[book V. 



Now, in the triangle ^^ 5 we have one side and the angles, 
to find the remaining parts. ^^ is found to be 28.88 chains 
and Bh to be 22.81 chains. We have now the complete field- 
notes of the area cut off. 



A 


S 60° E 


28.88 ch. 


B 


N 28f E 


22.81 


5 


N 57° W 


21.10 


6 


S 47° W 


25.00 



The area is found 
to be 58.5029 acres. 



It now remains to move this line northerly, so that the area 
contained between its present position and the new one shall be 
equal to 8.5029 acres. 

Suppose the lines A 6 and ^ 5 be prolonged till they meet at 
some point, as F, Fig. 114. 

Calculate A V and BV, also the area ABV. 
^F is found to be 92.19 chains and 5 F 88.18 
chains. The area of the triangle ABV, is 127.29 
acres. Let MN represent the line sought. 
Then, we have two similar triangles, with all 
the sides of the one, and the areas of each, 
known ; for, VMNmvi^t contain 8.5029 acres less 
than A VB. Then, AM and BN are easily 
determined. 

The complete notes of the area to be cut off, are 




FiQ. 114. 



M 


S 60° E 


27.89 


N 


N 28J° E 


19,82 


5 


N 57° W 


21.10 


6 


S 47° W 


21.87 



SEC. II.] PUBLIC LAI^^DS. 203 

Note. — Fields are so variously shaped that it is difficult to 
give rules that will apply to all cases. It is by practice alone 
that facility is obtained in that branch of surveying relating to 
the division of estates. We have given only a few examples 
that may serve as general guides in the application of the 
principles of Plane Geometry to such cases as may arise. 



SECTION II. 

PUBLIC LANDS OF THE UNITED STATES. 

225. Soon after the organization of the government, several 
of the States ceded to the United States large tracts of wild land, 
and these, together with the lands since acquired by treaty and 
purchase, constitute the public lands, or public domain. These 
lands were at first parceled out without reference to any general 
plan, in consequence of which the titles often conflicted with 
each other, and in many cases, several grants covered the same 
area. Through many years of labor and experiment, the present 
admirable system has been wrought out. 

J^26. This system is briefly that the territory to be surveyed 
shall be divided by true north and south, and east and west lines, 
into tracts, each of six miles square and containing as near as 
may be 23,040 acres, called Townships ; each township into 
thirty-six tracts, each of one mile square and containing as near 
as may be 640 acres, called Sections ; and each section into 
halves, quarters, and smaller portions, as may be deemed 
expedient. 

227. In the survey, all primary lines north and south are 



204 ELEMENTS OP SUKVEYIXG. [BOOK V. 

run on meridians of longitude, and all primary lines east and 
west are laid out on perpendiculars to the meridians. 

It is of the first importance, then, that meridians of longitude 
should be accurately determined, and that lines perpendicular to 
them should be determined with equal precision. 

The ordinary Surveyor's Compass is, for several reasons, not 
sufficiently accurate for the work of running out the standard 
and township lines, and such lines must be run by Burt^s Solar 
Compass, "or other instrument of equal utility." 

228. The Solar Compass for determining a true meridian 
was invented by William A. Burt, of Michigan, and patented by 
him in 1836. It has been improved, by him and others, from 
time to time since, and in its present state is represented and 
described in Appendix A. 

229. In commencing the division of the public lands in 
unsurveyed territory, an initial point is selected, with reference 
to its convenience in making the survey, perpetuated by a 
substantial monument suitably marked, and its true position in 
latitude and longitude determined. 

230. From the initial point the principal base line is run 
out due east and west with the solar compass, and permanently 
marked at each 40 chains, or half mile, with a quarter-section 
corner, and at each 80 chains, or mile, with a section corner. 

231. The principal meridian is then run out due north and 
south from the initial point, and also marked with monuments 
at intervals, like the base line. 

232. As meridians of longitude converge toward the poles, 
the distance between two such meridians decreases as the 
surveyor goes north. To counteract the error that would other- 
wise result from the convergency of meridians, and also to arrest 



SEC. II.] PUBLIC LAIRDS. 205 

error arisins^ from inaccuracies in measurements on meridian 
lines, standard parallels, or correction lines, are run east and 
west from the principal meridian and at stated intervals. On the 
north of the principal base-line, about latitude 35° north, these 
standard parallels are, in general, run at distances of e\erjfour 
townships, or twenty-four miles, and south of the principal base, 
at distances of every Jive townships, or thirty miles. Each of 
these standards is run out and marked in the same way a^ the 
principal base, and forms the base for laying out the townships 
north to the next standard parallel. The standards are num- 
bered according to their position with respect to the principal 
base-line, as 1st Standard Parallel South, 2d Standard Parallel 
South, 1st Standard Parallel North, &c. 

233. The principal meridian, the base-line, and the standard 
parallels having been first run, measured, and marked, and the 
corner boundaries thereon established, the exterior lines, of 
townships are then run, measured, and marked. 

The townships, consisting of a series of townships lying along 
a parallel, are numbered north or south of the principal base ; 
the first series north of the base being Township 1 North, the 
second series north being Township 2 North, &c. ; and the first 
series south being Township 1 South, &c.; these are designated 
T. IN., T. 2N., T. 1 S., &c. 

The ranges, consisting of tiers of townships, are numbered from 
the principal meridian both ways; the first tier west of the 
meridian being Range 1 West, the first tier east being Eange 1 
East, &c.; designated R. 1 W., R. 1 E., &c. 

234. The accompanying map, from the U. S. General Land 
Office, representing a considerable portion of the State of 
Arkansas, will serve for illustration. 

The principal meridian in this survey is called the 5th 



206 



ELEME2STS OF SUKVEYING. 



[book y. 




S location f^ Jj and Office 
"L Boundary of LandBisiricts 
o + Lands offered for Sale 



i.W.UKWK^O^OH .»^.H.'^! 



Fig. 115. 



SEC. II.] PUBLIC LANDS. 207 

meridian, and passes through the point of junction of the White 
river with the Mississippi. The principal base-line, running east 
and west, intersects this meridian a little to the east of White 
river; and from the meridian and base-line, reckoned from 
this point of intersection, all the ranges of townships are 
laid off. 

For example, 1 North, will apply to all the townships lying in 
the first row north of the base-line ; 1 South, will apply to all the 
townships in the first row south of the base line. Eange 1 East, will 
apply to all the townships lying in the first row, east of the 5th 
meridian ; and Range 1 West, will apply to all lying in the first row 
to the west of it. The small figures designate the rows of town- 
ships, reckoned north and south from the base-line, and the ranges 
reckoned east and west from the 5th meridian. Thus, Township 
1 North, Range 4 West, has its exact place designated, and may 
be immediately located. 

235. The diagram here given (Fig. 116), from the "Instruc- 
tions to the Surveyors-General of Public Lands of the United 
States," represents the required method of running out township 
lines. 

In the diagram, the upright figures (made thus, 1, 2, 3) 
commencing near the Principal Meridian and Base Line with 
No. 1, indicate the perambulations of the Surveyor in running 
the Townships and Correction lines. 

The Correction or Standard lines north of the base- are every 
four townships, and south of the base every five townships. 

The excess or deficiency of measurement on northern and 
southern boundaries is thrown on the westernmost half-mile. 

The measurements between meridian lines will, of course, 
always vary according to the latitude of the survey, besides 
being liable to be rendered inexact where the country is very 
hilly or broken. The convergency of the range lines as shown 



208 



ELEMENTS OF SUfiYEYING. 



[book V. 



by the measurements on this diagram, is according to calculation, 
as it exists between the parallels of 46° and 47° North Latitude. 

lHagram. 







r.x'ij 


ERIOKS OR i 

( 


TOrvVT^SHIP LINES. 










1 


p 
1 
1 










First 


correction or 


Standard Line 


ISTortli. 








80| 80 |B0 1 80 1 80] 80 


80|80 , 80, so 1 80 80 


80, 80 , 80] 80 1 80 ,80 


80, 30,80 ,80 


,80,80 




22 


K90 


80'8O' sol SO'OO 


769^ 


80 '80'80' SD'Sa 


i(i9b} 


80 '80' 80' 30 80 




80 '80 'so 


80 '80 


§ 


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11 


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oo 


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22 






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njij 




21 


© 


19 


o 


80 '80 80 80 '30 


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80 '80 '80 '80 '80 




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18 ' 


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16 


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80' 80 ' 80 ' 80 '80 


o 


80 ' 80 ' 80 ' 80 ' 80 




80 ' 80 ' so ' SO '80 




80 ' SO ' 80 


80 '80 


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80 ,80 ,80 ,80,80 


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80 18O1 80,80,80 


80 


80 ,80 ,80 


80 ,80 


© 



Base 



Line 



Fig. 116. 



236. The "Instructions to the Surveyors-General " for run- 
ning, measuring, and marking the exterior lines of townships, 
are as follows: 



SEC. II.] PUBLIC LANDS. 209 

Townships situated North of the Base Line and West of the 

Principal Meridian. 

Commence at No. 1 (see figures on the diagram, Fig. 116), 
being the southwest corner of T. 1 N. — K. 1 W., as established on 
the base line ; thence north on a true meridian line, four hun- 
dred and eighty chains (6 miles), establishing the section and 
quarter-section corners thereon, as per instructions, to No. 2, 
whereat establish the corners of Tps. 1 and 2 N. — Es. 1 and 2 W.; 
thence east, on a random or trial line, setting temporary section 
and quarter-section stakes to No. 3, where measure and note the 
distance at which the line intersects the eastern boundary, north 
or south of the true or established corner. Run and measure 
westward, on the true line (taking care to note all the land and 
water crossings, &c., as per instructions), to No. 4, Avhich is 
identical with No. 2, establishing the section and quarter-section 
permanent corners on said line. Thence proceed in a similar 
manner from No. 4 to No. 5, No. 5 to No. 6, No. 6 to No. 7, and 
so on to No. 10, the southwest corner of T. 4 N. — E. 1 W. 
Thence north, still on a true meridian line, establishing the 
mile and half mile corners, until reaching the Standard Par- 
allel or correction line ; throwing the excess over, or deficiency 
under four hundred and eighty chains, on the last half mile, 
according to law, and at the intersection establishing the 
" Closing Corner," the distance of which from the standard 
corner must be measured and noted as required by the instruc- 
tions. But should it ever so happen that some impassable bar- 
rier will have prevented or delayed the extension of the standard 
parallel along and above the field of the present survey, then 
the deputy will plant, in place, the corner for the township, 
subject to correction thereafter, should such parallel be ex- 
tended. 



210 



ELEMEKTS OF SURVEY1N"G. 



[book V. 



North of the Base Line and East of the Principal Meridian. 

Commence at No. 1, being the southeast corner of T. IN. — 
R. IE., and proceed as with townships situated "north and 
west," except that the random or trial lines will be run and 
measured ivest, and the true lines east, throwing the excess over 
or deficiency under four hundred and eighty chains on the west 
end of the line, as required by law ; wherefore the surveyor will 
commence his measurement with the length of the deficient 
or excessive half-section boundary on the west of the township, 
and thus the remaining measurements will all be even miles and 
half miles. 

237. In running random township exteriors, if such random 
lines fall short or over-run in length, or intersect the eastern or 
western boundary, as the case may be, of the township, at more 
than three chains and fifty links north or south of the true 
corner, the lines must be retraced, even if found necessary to 
re-measure the meridional boundaries of the township. 

238. The exterior lines of townships having been established 
and duly marked, each township is divided into 36 squares, 
called Sections, by meridians one mile apart, and by east and 
west lines at the same distance from each other. The sections 
of a township are numbered from 
1 to 36, beginning at the north- 
east angle and proceeding as shown 
in the annexed diagram : 

To describe a section accurately, 
we say, for example, section num- 
ber 5, in township number 4 north, 
in range 3 west of a known 
meridian. 













\ 


6 


5 


4: 


3 


2 


1-- 


7 


8 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


23 


24 


30 


29 


28 


27 


26 


25 


31 


32 


33 


34 


35 


36 



239. The sections are divided 



Fig. 117. 



w 



SEC. II.] PUBLIC LANDS. 211 

into half -sections, quarter-sections, and even into eighths of 
sections. The following table shows the contents of a township 
and its subdiyisions : 

1 township = 36 sections = 23040 acres. 

1 section = 640 acres. 

^ section = 320 acres. 

J section =160 acres. 

-J section = 80 acres. 

240. As any excess or deficiency of measurement is, by law, 
to be thrown on the extreme tier of sections and half-sections 
contiguous to the north and west boundaries of townships, such 
sections and half-sections are sold as containing only the quan- 
tity expressed in the returns and plots, respectively, and all 
others as containing the complete legal quantity. 

The government surveyors are rarely required to subdivide a 
section into quarters, as that work properly belongs to the 
county surveyors. 

Note. — The student is referred, for more detailed information on 
Government Surveys, to : Instructions to tlie Surveyors-General of Public 
Lands of the United States, prescribed by the Commissioner of the General 
Land Office ; Clevenger's Government Sui-veying ; Burt's Solar Compass. 



BOOK VI. 

TRIGONOMETRICAL SURVEYING. 



SECTION I. 

MAKING THE SU RVEY. 

241. Trigonometrical Surveying, or Triangulation, is 

the method of determining the position of points on the surface 
of the earth by the apphcation of the principles of Trigonometry 
— each successive point being determined by the intersection of 
two lines which make known angles with a given line. It may 
be used in small or extensive surveys ; it does not necessarily 
take into account the curvature of the earth, though always used 
in the great surveys, such as the U. S. Coast Survey, in which 
that is considered. 

242.* In the construction of a true map of any large 
territory, three things are necessary. The first of these is to 
ascertain the exact relation to each other, as to distance and 
direction, of the leading features of the country ; selecting such 
high points as may be seen from the greatest distance, and con- 
necting the whole by a chain of triangles of the largest dimen- 
sions practicable, and of the forms most convenient for accuracy 
of computation. 

The series must rest upon a base of which the geographical 
position has been established with mathematical accuracy. 
The points which form the several angles of the chain will thus 

* Report of N. Y. State Survey, 1878. 



SEC. I.] MAKING THE SURVEY. 213 

be fixed with equal exactness as points of the general surface of 
the earth. With the sides of this principal chain of triangles 
as bases, a net of smaller triangles is then to be constructed 
occupying the interior of the larger, and resting their angles 
upon the most conspicuous objects observable from the principal 
stations. And within these still a third series is to be formed, 
connecting as many of the less important points within those of 
the second order as the objects of the survey may seem to require. 
Every angle of every series of these triangles marks a geograph- 
ical point determined with the same degree of precision as that 
of the original base from which the triangulation began. The 
base on which the New York State Survey rests is a side 
23 miles long of one of the large triangles of the United States 
Coast Survey, a magnificent work which twenty years since 
pushed its operations to the head of tide-water upon the Hudson 
Eiver. The bases of the United States Coast Survey itself are 
lines of some miles in length, determined in position by astro- 
nomical observation, and actually measured on the ground from 
end to end, by means of apparatus of extreme delicacy, con- 
structed especially for the purpose. 

243. The specimen of triangulation. Fig. 118, is from a 
survey, made by Professor Eees of Columbia College, of Otsego 
Lake, N. Y., for the N. Y. State Survey, and will serve to illus- 
trate the method of making and plotting such a survey. 

244. Before commencing a trigonometrical survey, an ex- 
amination of the entire territory should be made for the purpose 
of selecting a base line and proper points for stations ; this 
examination should be more or less elaborate according to the 
nature and extent of the survey. 

The proper distribution and combination of the triangles, so 
as to adapt them to the survey in hand, require great judgment 
and care, and the selection of proper trigonometrical points is a 



u 








i^ 








< 








J 













> 




o 




> 




hi 




DC 










a> 


CO 


X 


UJ 


h 




1- 
e: 
O 




ir 




Q 


< 




u. 


< 


h 


O 





co 


0. 


>- 




> 




u 




z 




> 








X 








D 








CO 













SEC. I.] MAKING THE SrKYEY. 215 

very important part of the preliminary operations. The selec- 
tion should be so made that each point may command a view of 
the greatest practicable number of surrounding trigonometrical 
point objects, and that each angle at the point shall be as near as 
may be 60°. A triangle which has an obtuse or a very acute 
angle will experience a greater change of form, for a given error, 
than one which is nearly equilateral ; and since the accuracy of 
each triangle depends upon the preceding ones, it is evident that 
the introduction of a single "ill-conditioned" triangle might 
vitiate the whole survey. No angle less than 30° or more than 
120° should be used; and even such angles should not 
be admitted when the locality can be so chosen as to pre- 
vent it, 

245. If the tri angulation is to be over a limited extent of 
country which has already been covered by a net-work of 
Primary Triangles, a side of one of these triangles should be 
used as a base. It is never good practice to measure a base line, 
when a side of a triangle of a previous survey is available ; but if 
no such side can be obtained, then the selection of a proper site 
for a base-line forms one of the first objects of the preliminary 
reconnaissance. It should, if possible, be fixed on an open plain, 
free from surface encumbrance or freed from such. It must be 
so chosen that the surrounding signals may be distinctly seen 
from its extreme points ; and hence those signals which mark 
points of the adjacent triangulation, should be selected with 
reference to the base. The length of the base should be suited 
to the magnitude of the survey. 

246. In measuring a base-line, every possible precaution 
should be taken to insure accuracy. The line measured should 
be straight, to effect which it should be ranged out with the 
transit. The ends of the base should be marked by a stone sunk 
in the ground, with a copper bolt let into it and the exact point 



216 



ELEMENTS OF SURVEYING. 



[book VI. 



of beginning and ending fixed by the intersection of two lines 
cut into the head of the bolt. 

The measurement may be made with steel tape or rods. If a 
tape is used, it should be carefully drawn out each time to its 
standard length, and should be compared with a standard both 
before and after measurement, and correction made for its 
variation, if any, from standard. The mean of several measure- 
ments should be taken for the correct measurement. If the 
measurement has been made on an incline, instead of on a level, 
the measured distance should be reduced to the horizontal 
distance by multiplying the inclined distance measured by the 
cosine of the angle of inclination. 

247. For a description of the base-line apparatus used in the 
U. S. Coast and Geodetic Survey, see Reports of that Survey for 
1854, 1857, and 1880. 

248. The alignment of the measuring tape, or rods, both 
vertically and horizontally, or in the line of the slope if the 
measurement be not horizontal, is of the greatest importance, 
since there is no compensation of errors, a faulty alignment 
always resulting in a measured length greater than the true 
length. 

249. Having carefully measured the 
base, it is then necessary to reduce the 
measurement to the sea level. 

Let L = measured base. 
Let I = reduced base. 
Let R = radius of earth. 
Let h = average height of measured 
base above sea level ; tia. ng. 




then, 



I : L :: R : R+h; 



SEC. I.] MAKING THE SURVEY. 217 

whence, l=^ = L (-^|^) = L (l-^'^ • 

But h is so small as compared with R that we may, without 
sensible error, make R+h = R; whence we have, 

250. The trigonometrical stations are marked by signals, 
which may be constructed in a great variety of ways, depending 
upon the locality of the stations and the lengths of the sides of 
the triangles. 

Sometimes a signal has to be raised above the level of the 
adjacent country, in which case it is constructed of timbers, and 
upon the apex is placed a vertical staff bearing a flag. The 
exact trigonometrical point is determined by a plumb-line 
suspended from the apex of the signal. 

A temporary signal may be constructed with 
three or four pieces of scantling framed and 
braced, as shown in the annexed figure, with a 
short pole projecting from the apex. The plumb 
determines the point B, which is the exact 
trigonometrical point over which the theodolite is L, 

to be placed. Where the sides of the triangles // 
are not very great, a pole, planted vertically and d 
surmounted by a flag, will answer as a signal. ^^^' ^^• 

In order to distinguish the different signals, the flags which 
they bear should be different from each other. They may be 
formed by arranging stripes of white and red, according to some 
prearranged plan, aud the flags of the different stations should 
be entered in a book. For the purpose of future reference, 
the trigonometrical point at each station, as B, should be indi- 



218 ELEMENTS OE SURVEYING. [bOOK VI. 

cated by a permanent mark. If the point falls upon a rock, a 
hole may be drilled to show the locality ; or if not, a mark-stone 
may be sunk under the point, deep enough to be beyond the 
reach of accident. A record of the monument should be pre- 
served, together with its reference to some of the permanent 
objects in the neighborhooa, 

251. A Heliotrope is necessary in long sights, and is always 
of great service in short sights in directing the observer to the 
station-mark. It consists essentially of a small mirror, so 
directed by an assistant as to throw a beam of sunlight into the 
telescope of the distant observer. Let a silvered glass, about 3 
inches square, be mounted on a board in a manner similar to the 
telescope mount of a transit so as to have a motion about a 
horizontal axis and at the same time about a vertical axis ; in 
front of this, at a distance of two or three feet, mount a board 
with a hole in it, across which hole threads are to be stretched 
at right angles to each other, and adjust this hole over the station 
by a plumb-line. At the centre of the back of the mirror scrape 
away the silver, making a small sight-hole ; if now an assistant, 
sighting through the hole in the mirror, moves it so that the 
cross threads come in line with the distant station, it will be easy 
to keep the beam upon the observer by properly inclining and 
revolving the mirror. As the light reflected from so large a 
mirror would be too intense to observe with the telescope, it is 
necessary to make the cross-thread hole in the board quite small, 
not more than |- inch diameter for distances not exceeding 5 
miles, about one inch diameter for distances of ten miles, and 
so on. 

A small pocket mirror will be found very useful as a means 
of telegraphing instructions, by combinations of flashes according 
to a system previously agreed upon. It can be directed to the 
observing station (or to the observed station) with sufficient 



6EC, I.] MAKING THE SURVEY. 219 

accuracy for signalling, by setting a vertical staff in line with the 
distant station and causing flashes to travel up and down the 
staff. 

252. The extent of the survey, and the standard of accuracji 
to which the results are required to conform, must determine the 
size and perfection of the instrument to be employed in the 
measurement of angles. The angles of the primary triangles of 
the United States Coast and Geodetic Survey are measured with 
theodolites, whose horizontal circles are 24 or 30 inches in 
diameter; and to eliminate, as much as possible, every source of 
error, great numbers of observations are made at each station, 
the readings being made on different points of the arc by different 
verniers. Usually from 40 to 60 observations are made for each 
angle — one measurement, with the telescope direct, and one with 
it reverted, constituting a complete observation. With these 
precautions, it has been found that the error in a primary 
triangle (where the sum of its three angles has been compared 
with 180°), has fallen much within 3 seconds. The error of 
3 seconds has been adopted as the highest admissible limit of 
error in such triangles. 

253. The theodolite does not differ essentially in the 
principles of its construction and use from the transit, which has 
already been described. It is fitted with many appliances, for 
accuracy in the observation and the reading of angles, which it 
is unnecessary to describe here. Fig. 121 is a representation of 
the 8 to 12 inch theodolite used in the U. S. Coast and Geodetic 
Survey, taken from the Report of that Survey for 1880. 



2W 



ELEMENTS OF SURVEYIXG. [BOOK VI. 




Fig. 181. 



SEC. I.] 



MAKING THE SURVEY. 



221 




Fig. 122. 

254. To illustrate the principle of repetition in ttie measure- 
ment of angles, suppose the of the vernier to coincide with the 
of the limb, and the telescope to be directed, from the station 
A, Fig. 122, upon one of the objects, as the signal at B. 
Clamp the limb and, unclamping the vernier plate, direct the 
telescope on the second object, as the signal at E. If we now 
clamp the vernier plate and, unclamping the limb, direct the 
telescope on the signal at B, the line (0°, 180°) of the limb will 
make with AB 'du angle equal to BAE. Again clamp the 
limb and, unclamping the vernier plate, direct the telescope 
on the signal at E. The reading will evidently be equal to 
twice the angle BAE; and if we repeat the operation, the 
reading will be three times the angle, and so on. After ten 
repetitions, if we add 360° each time the of the vernier 
passes the of the limb, the final reading will be ten times the 



li 



222 ELEMEl^TS OF SURVEYING. [bOOK VI. 

angle BAB, affected with the joint errors of the ten observations, 
and one- tenth of this will be the reading required to a greater 
degree of accuracy than could probably be attained by a single 
observation. 

255. The method of reading angles and recording notes is 
as follows: 

(1.) First mark one vernier A and the other B by pasting 
these letters upon the vernier plate. Set vernier A at 0° and 
clamp it. Direct the cross-wires to the left of station B and 
then by a careful movement to the right bring the sight 
nearly upon station B, heing careful not to pass the station, 
and perfect the bisection by the lower tangent screw. Read 
both verniers, reading B to minutes and seconds only, and 
make the entry as in the table subjoined. Now loosen 
the vernier clamp and turn carefully to the right till nearly 
upon station E, completing the bisection by the vernier 
tangent-screw. Now make a " check " mark in column 
"vernier ^," as in table. Next loosen the lower clamp and turn 
the telescope to the right till station B is nearly reached, and 
repeat the previous operations, making *^ check" in "vernier A " 
column as before. In like manner repeat the operations for the 
third time, and enter the final reading upon station E, as in the 
table. 

If the motion of vernier A has h^en forward, in the direction 
of the graduation of the limb, the first reading must be marked 
negative, as in the table ; when the motion of vernier A is back- 
ward, the final reading must be marked negative. In the case 
just described the motion has been forward, therefore from the 
mean of the last readings (217° 15' 15"), we subtract the mean 
of the first readings (0' 10"), and divide the result by the 
number of repetitions (3) obtained for the mean angle 72° 
25' 1".7, 



SEC. I.] . MAKING THE SURVEY. 223 

(2.) Set vernier A at 180°, and having turned the telescope 
to the right of E, make three repetitious of the angle subtended 
by BB, working from right to left. In this case vernier A will 
pass over the 360° mark, which fact is noted in the table under 
the first "vernier ^" reading. As vernier A worked backward, 
the last reading, that taken upon B, must be negative. 

(3.) Kevolve the telescope upon its horizontal axis, and turn 
the telescope about the vertical axis towards B. Set vernier A 
at 90° and repeat three times from B to E. 

(4.) Set vernier A at 270° and repeat three times from 
B to B. 

The final mean of the four angles is 72° 24' 59". 2. 

Both verniers are read to eliminate error of centering of 
graduated limb. 

Kepetitions are made to eliminate errors of graduation. 

Eeadings from left to right, and from right to left, are taken 
to counteract errors due to torsion, and personal error of 
bisection. 

The telescope is reversed on its horizontal axis to eliminate 
error due to inequality of axis points, and error of colli mation. 



224 



ELEMENTS OF SUEYEYING. 



[book VI. 



Angles taken at Station A, July 17, 1883. 



Sta. 
Obs. 

B 


Vernier A. 


B. 


± 


Calculation. 


Mean Ang. 


0° 0' 0" 


0' 20" 


— 


217° 15' 15" 




E 


217° 15' 10" 


15' 20" 




10 
3)217 15 5 


72° 25' 1".7 


E 


180° 0' 0" 


0' 10" 




180° 0' 5" 
360 






v/f/v/ 


360° 




540 5 
322 45 15 


• 


B 


322° 45' 10" 


45' 20" 


— 


3)217 14 50 


72° 24' 56".7 


B 


90° 0' 0" 


0' 0" 


— 


307° 14' 50" 




E 


307° 14' 40" 


15' 0" 




90 
3)217 14 50 


72° 24' 56".7 


E 


270° 0' 0" 


0' 20" 




270° 0' 10" 




B 


52° 45' 0" 


45' 10" 




52 45 5 
3)217 15 5 


72° 25' 1".7 












4) 99' 56".8 










Final mean. 


72° 24' 59".2 



SEC. I.] 



MAKING THE SURVEY. 



225 



256. If several stations can be read in succession, a simple 
modification of the method is practicable. Suppose the observing 
station to be E of Fig. 122, and we wish to read A, H, B, (7, &c. 
Sight A with the vernier set at 0° ; unclamp the vernier and 
turn to the right till ^Tis bisected and enter the reading opposite 
H in the table below ; unclamp the vernier and sight B, and 
enter its reading; continue the motion to the right, reading each 
station in turn. Suppose to be the station on the extreme 
right ; now set the vernier at 180° and work from (7, around to 
the left, to A, entering the readings in the table in a reverse 
order, from the bottom to the top of the page. Eeverse the 
telescope, and with 90° as the first reading work from A to C, 
and then with 270° as the first reading work from (7 to A. The 
notes are given below ; the readings on station A subtracted from 
the corresponding readings on H, B, and G, give the angular 
distances of these points from the line EA. 

Angles taken at Station E. 



Sta. 


Vernier A. 


B. 


Mean Read. 


Angles 


with Line 

EA. 


Mean Angles with 
EA. 


A 


0° 


0' 0' 


0' 20" 


0° 


0' 10" 










72 


10 





72 


5 










90 





10 


90 


5 










162 


20 


10 


162 


15 








H 


40° 


0' 10" 


0' 20" 


40° 


0' 15" 


40° 


0' 5" 






112 





10 


112 


5 


40 









130 


10 


30 


130 


20 


40 


15 






202 


10 


10 


202 


10 


39 


59 55 


40° 0' 3".75 


B 


66" 


30' 0" 


30' 20" 


66° 


30' 10" 


66° 


30' 0" 






138 


29 40 


30 20 


138 


30 


66 


29 55 






156 


29 40 


30 10 


156 


29 55 


QQ 


29 50 






228 


30 


30 


228 


30 


66 


29 45 


66° 29' 52".5 



Z2b 



ELEMENTS OF SURYETiyG. 



[boos YI. 



Sta. 


Vernier A. 


B. 


Mean 


Read. 


Angles with Line 
EA. 


Mean Ar>sles with 
EA. 


c 


108" 0' 0" 


0' 10" 


108° 


0' 5" 


107° 59' 55" 






180 


20 


180 


10 


108 5 






198 10 


10 


198 


10 


108 5 






270 


10 


2:0 


5 


107 59 50 


107°57'58".75 



Having found by these methods the angles of any triangle, 
one of its sides being the base-line or a kno^m side of a triangle 
already computed, we can find the sides of the new triangle. 

First subtract the sum of the angles from 180° and apply ^ of 
the error to each angle; next treat the triangle as a plane 
triangle and compute the two required sides. 

257. The spherical excess in triangles of a Primary System 
is seldom more than about 6", and triangles whose sides are not 
more than ten miles long may be regarded as plane triangles 
without sensible error. 



258. It sometimes happens that a steeple, tower, or other 
prominent object must be used as a station, and in most cases it 
is impossible to set the instrument over the 
centre of such station. In such cases a 
"reduction to the centre" is necessary. 

Let be the position of the instrument, 
C the centre of a circular tower which marks 
the station, and DCG the desired angle, D 
being the right-hand object and G the left- 
hand one. Measure (in some one of the 
many ways for indirect measurement) the 
distance OC^ r, and take the angle GOC= y, 
always measuring the angle y (called the 
angle of direction) from the left-hand object, iig. 123. 




SEC. T.] MAKING THE SURVEY. 227 

Gy and estimating it towards the left, from 0'^ to 300", as in the 
figure. The angle 1 being exterior to the triangle DOl we have 
I =z 0-{-a, and from the tnangle GIC we have /— C -\- (3, 
from which we deduce 

C=0-\-(a-fi) (1) 

We now need to determine the correction {a — /3). In 

the triangle JJOC we have (calling the side DC, D) sin a := 

r ^\n [0 -\- y) , . , __. • • -• i n 
-^— ^— (Art. 32), or as a is, m practice, always very small 

we may substitute for its sine its value in seconds, making sin 

a = (a)'' sin 1'', which substituted above skives (a)"=—,--i — rrr^* 
\ / ° ^ ^ /> sm 1 

In like manner (calling the side GO, G) we have (/3)" = 

7* sm 11 

-jr—. — 4n' Substituting? these values in formula (1), we obtain 
G sin 1 ° ' 

( D sin 1 6^ sin 1 ) ^ ^ 

This formula is general if care is taken to estimate the angle 
y with reference to the positions of the stations D, G, and C, as 
above directed, the signs of the trigonometric functions also 
being observed (see Davies' Leg., Trig. Art. 58). 

The angle is measured in the usual manner. The angle y, 
in the case supposed, is obtained by measuring the angle which 
each tangent to the tower through makes with the line OG, 
and taking their half sum. 

In formula (2) the sides D and G are unknown. In triangles 
whose sides are not less than 3000 feet, r being relatively very 
small, it will be sufficient to calculate these sides from the known 
side and the uncorrected observed angles, and substitute such 
computed values in formula (2). 

When the sides are less than 3000 feet, r being relatively 
large, the values of D and G are obtained by two approximations. 
First plot a triangle 0' D' G' upon as large scale as practicable, 



228 ELEMElfTS OF SUEVEYIKG. [BOOK VI. 

using the known side and the observed angles; then plot the 

angle y, and lay off O'C = nr, n being either 10, 20, 30, &c., so 

that nr may measure between -^ and ^ of the length of the 

shorter of the two plotted sides O'U and 0' G\ in which case 

0' D' C 
the plotted angle O'D'C will be equal to ^x«, or a = , 

0' G' C ' 
and (i = These values substituted in formula (1) 

give a value of C which may be used in the computation of the 
sides D and G, and these computed values of D and G must then 
be substituted in formula (2). 

259. An error of observation may also arise from the 
unequal illumination of the face of the object presented to the 
observer, called error due to phase ; thus, if an 

observer at A sight a tower at D, the sun being 
m the direction S, he will direct his telescope to 
dj the middle of the illuminated portion, instead 
of to D. This error, and also that due to 
irradiation in the case of ordinary signals, may 
be avoided by observing when the sun is 
obscured, or by throwing a shadow upon rods and 
other commonly used signals. It is hardly worth while to intro- 
duce a formula for correction of phase in this connection. 

(For the methods used in the work of the U. S. Coast and 
Geodetic Survey, see "Field Work of the Triangulation," 1877, 
by Richard D. Cutts, assistant.) 

260. When, in connection with a trigonometrical survey on 
shore, a harbor is to be surveyed (see Fig. 122), for the purpose 
of ascertaining the channels, their depth and width, the positions 
of shoals, and the depth of water thereon, other means must be 
used, and other examinations made, in addition to those already 
described, 




If 



SEC. I.] MAKII^G THE SURVEY. 239 

Let buoys be anchored on the principal shoals and along the 
edges of the channel ; and using any one of the lines already 
determined as a base, let the angles subtended by lines drawn 
from its extremities, to the buoys respectively, be measured with 
the theodolite. Then, there will be known, in each triangle, the 
base and the angles at the base, from which the distances to the 
buoys are easily found ; and hence, their positions become 
known. 

Having made the soundings and ascertained the exact depth 
of the water at each of the buoys, several points of the harbor 
are established, at which the precise depth of the water is 
known ; and by increasing the number of the buoys, the depth 
of the water can be found at as many points as may be deemed 
necessary. 

261. If a person with a theodolite, or transit, be stationed at 
each extremity of the base-line, it will not be necessary to 
establish buoys. A boat, provided with an anchor, a sounding- 
line, and. a signal-flag, has only to throw the anchor, hoist the 
signal-flag, and make the sounding, while the persons at the 
extremities of the base-line measure the angles. From these 
data, the precise place of the boat can be determined. 

262. There is another method of determining the places at 
which the soundings are made, that admits of great despatch, 
and which, if the observations are made with care, affords results 
sufficiently accurate. 

Having established, trigonometrically, three points which can 
be seen from all parts of the harbor, and having provided a 
sextant, let the sounding be made at any place in the harbor, and 
at the same time the three angles subtended by lines drawn to 
the three fixed points, measured with the sextant. 

The problem, to find, from these data, the place of the boat 
at the time of the sounding, is the same as Example 6, Art. 203. 



230 ELEMENTS OF SURVEYIN'G. [BOOK VI. 

It is only necessary to measure two of the angles, but it is 
safest to measure the third also, as it affords a verification of 
the work. 

The great rapidity with which angles can be measured with 
the sextant by one skilled in its use, renders this a most expe- 
ditious method of sounding and surveying a harbor. 

For description of the sextant and its method of use, see 
Appendix B. 

263. There is yet another method of finding the soundings, 
which, although not as accurate as those already explained, 
will, nevertheless, afford results approximating nearly to the 
truth. It is this: — Let a boat be rowed, with uniform speed, 
across the harbor, from one extremity to the other of any of the 
lines determined trigonometrically. Let soundings be made 
continually, and let the precise time of making each be carefully 
noted. Then, knowing the length of the entire line, the time 
spent in passing over it, as also the time of making each of the 
soundings, we can easily find the points of the line at which the 
several soundings were made ; and hence, the depth of water at 
those points becomes known. 

264. If a person stationed on shore with a theodolite or 
transit, takes the bearing of the boat, at every second or third 
sounding, determined by hoisting a flag, it will fix the positions 
of the soundings with great accuracy. Soundings may thus be 
made along any number of known lines, and a comparison of 
the depths found, on different lines, at or near their points of 
intersection, will show with what degree of accuracy the work 
has been done. 

Sounding-lines should be made of strong cord, and divided 
into feet or fathoms, by different colored rags or other marks. 
The lead is shaped like the frustum of a cone, with the base B 
hollowed out, to hold some grease. The land or mud of the 



SEC. I.J 



MAKING THE SURVEY. 



231 




Fig. 125. 



bottom adheres to the grease, and thus shows the nature of the 
bottom, which should be entered in the field- 
book, and laid down upon the map. As the 
cord is liable to change its length, it should 
be compared, from time to time, with some 
standard. In tide-waters, the exact time of 
each sounding is to be noticed, and an 
assistant should note the height of the tide at 
regular intervals, upon a tide-gauge. The 
tide gauge is permanently placed at some 
convenient point of the harbor, and its 
point is referred, by means of a spirit-level, to some fixed bench- 
mark, on a level with mean low-water mark, to which all the 
soundings must be reduced. 

265. Having plotted the work done with the theodolite or 
transit, as also the outline of the harbor traced with the compass, 
it remains to delineate the bottom of the harbor ; and this is done 
by means of horizontal curves, hereafter explained (Art. 332), 
which are used to represent broken or undulating ground. 

Let the plane of reference be taken through low-water mark, 
or to coincide with the surface of the water at low tide. The 
accuracy with which the bottom of the harbor is to be delineated, 
will guide us in fixing the distance between the horizontal planes 
of section. 

The first horizontal plane should be passed ac a aistance 
below the shallowest point that has been sounded, equal to the 
number of feet fixed upon for the distance between the planes 
of section ; and the curve, in which it intersects the bottom of 
the harbor, determined as in Art. 335. And similarly, for the 
other horizontal planes of section. 

Having thus delineated the bottom of the harbor, and noted 
on the map the distance of each intersecting plane below the 



232 ELEMEN'TS OF SURVEYING. [BOOK VI. 

plane of reference, let such lines be drawn as will indicate the 
channels, shoals, sunken rocks, and direction of the current. 

In the example given in Fig. 122, soundings have been 
made in three directions, from the sand-bar in the harbor, and 
also from the rocky shore across to the light-house. 

266. When a large extent of territory, or a long line of sea- 
coast is to be surveyed, it becomes necessary to consider the 
curvature of the earth's surface ; this branch of surveying is 
called Geodetic surveying. 

The operations necessary to the successful execution of a 
Geodetic Survey, require the minutest attention, and when 
performed, numerous corrections are to be applied to the meas- 
ured lines and angles, on account of the various causes of error 
incident to such operations. 

To investigate those causes of error, and to deduce rules for 
correcting the errors, in all cases, would exceed the limits of 
this treatise. We have, therefore, attempted nothing more 
than an outline of the operations in a trigonometrical survey, 
in which the Plane-Table and Compass are used in con- 
nection with the Theodolite or Transit, and in which the 
curvature of the earth is not considered. 



SEC. II.] 



FILLIis^G UP THE SUEVEY. 



233 



SECTION II. 

FILL! NG U P TH E SU RVEY. 

After the triangulation is completed, the interior may be 
filled up by the aid of the Compass, or Plane-Table, or both. 

By the Compass. 

267. The use of the compass, in determining points and lines, 
by means of offsets, has been already explained (Art. 123). We 
will apply these principles in the example of the harbor, Fig. 122. 

Place the compass at A., and take the bearing of the line 
AE, which is S 12° W. 




Enter this bearing at A. Then measure along the line AH 
any distance, sls Aa equal to 130 yards, and make an offset to 
the lake, which we measure and find to be 50 yards. Enter 
the 130 in the middle column, and as the lake lies on the right 



234 ELEMENTS OF SURVEYIl^G. [bOOK VI. 

(in going from A to E), we insert the 50 in the right-hand 
column. 

We then measure along the line AB to b, 350 yards from A. 
Here we make a second offset to the lake, and find it to be equal 
to 100 vards. Having entered the distances in the notes, we 
measure to q, the point where the line AF touches the creek, 
and we enter the distance from A, 415 yards. 

At d, we lay off an offset on the left, to the pond, 70 yards ; 
at e, an offset to the mouth of the creek, 150 yards ; and at E, 
where the course terminates, an offset to the lake, of 160 yards. 
The entire distance from A to E is 800 yards. 

At E, we take the bearing to H, which is N 50° E. Haying 
measured along this line to /, 315 yards, we make an offset to 
the pond, on the left, of 50 yards, and to the shore, on the right, 
of 90 yards. Having entered these distances, we recommence 
the notes at 315, below, which we suppose to be at the bottom 
of the second page. Having reached ff, the extremity of the 
course, we enter the entire distance from E, 680 yards. We 
next take the bearing to /, S 52° E. We then measure the dis- 
tances to m, n, p, and /, and enter them, together with the 
offsets, as in tlie notes. 

It is also well to make, in the columns on the right and left, 
such sketches of the ground, fields, houses, creeks, and rivers, 
as will afford the means of making an accurate delineation on 
paper. 

By the Plane-Table. 

268. The plane-table. Fig. 127, consists of two parts: a 
rectangular board, and a tripod to which it is firmly secured. 

Directly under the rectangular board are four milled screws 
which pass through sockets inserted in a horizontal brass-plate ; 
these screws are worked against a second horizontal plate, for 



SEC. 1I.J 



FILLING UP THE SURVEY. 



235 



the purpose of leveling the table ; the table having a ball-and- 
socket motion, similar to the limb of the transit. 




Fig. 12T. 

Between the upper horizontal plate and the table, there Is 
a clamp-screw, similar to the clamp-screw of the transit, which 
being loosened, the table can be turned freely about its axis. 
There is, also, a small tangent screw, by which the smaller 
motions of the table are regulated, after the clamp-screw is 
made fast. 

A set of brass clamps for fastening the paper to the table ; 
a clamp with an attachment on one side for fastening a plumb- 
line, and, on the other, a pin immediately over the point of 
attachment of the plumb-line, for marking on the paper a point 



236 ELEMEifTS OF SURVEYING. [BOOK VI. 

directly over a station of the ground ; an Alidade and a Declina- 
tor accompany the Plane-Table. 

The Alidade is a brass ruler, from one to two feet long, 
with a fiducial or true straight edge. On the upper face of 
the ruler are two spirit-levels at right angles to each other, 
and, near the middle of the ruler, a brass standard w^hich carries 
a telescope, like the telescope of the transit, with a vertical 
arc for measuring vertical angles, and micrometer wires for 
measuring distances (see Art. 206). The telescope is so placed 
with regard to the edge of the ruler beneath, that its line of 
collimation, when properly adjusted, is parallel to that edge and 
in the same vertical plane. 

The Declinator is a metal box, containing a magnetic needle 
with a range of about 10° on each side of the 0. It is used to 
orient the table, i. e., to place it at any point of the field in the 
same position with respect to the points of the compass that it 
had at other stations in the same survey ; or, in other words, to 
place it in such position at a new station that a line previously 
drawn upon the paper shall be parallel to the line of sight which 
it represents. The orientation may be effected as folloAvs : 
While the table is in position for drawing the first lines of 
the survey, place upon it the declinator, with one of the longer 
edges of its box set along a line drawn on the paper for the 
purpose, and note the reading of the needle ; when the table is 
removed to a new position, turn it till the declinator, placed 
along the line as before, gives the same reading. Figure 127 
represents the plane-table and its accompaniments as used on 
U. S. Coast Survey. 

269. The plane-table is used to determine the shorter lines 
of a survey in extent and position. 

Having placed a paper on the table, examine the objects and 
lines which are to be determined, and select for a base such a 



IP 

SEC. II.] FILLING UP THE SURVEY. 237 

line of the triangulation that most of the objects can be seen 
from its extremities. Then place the plane-table over one ex- 
tremity of the base ; make it truly horizontal, and turn it until 
the larger part of the paper lies on the same side of the base 
with the objects. 

Then, tighten the clamp-screw, and mark with a pointed 
pin the point of the paper directly over the station, which point 
is determined most accurately by suspending a plumb from 
the lower side of the table. Press the pin firmly on this 
point, bring the fiducial edge of the ruler against it, and sight 
to the other extremity of the base-line, and mark, with the pin 
or pencil, the direction of the line on the paper. Sight, in 
like manner, to every other object, and draw on the paper 
the corresponding lines, numbering them from the base-line, 
1, 2, 3, 4, &c. 

Then, with a pair of dividers, take from the scale a certain 
number of equal parts, to represent the base, and lay off the 
distance on the base-line from the place of the pin. Take up 
the table, carry it to the other extremity of the base, and 
place the point of the paper corresponding to that extremity, 
directly over it. Place the fiducial edge of the ruler on the 
base-line, and turn the table, by means of the tangent screw, 
until the sights are directed to the first station. If, however, 
in bringing the table to this position, the corresponding point 
of the paper has been moved from over the extremity of the 
base-hne, move the legs of the tripod until it is brought 
back to its place. Let the table be then leveled, after which, 
place the ruler again on the base-line, and bring the table 
to its proper position, by the tangent-screw, and continue the 
adjustment until the extremity of the base-line, on the paper, 
is directly over the station, and in the same vertical plane 
with the base-line, on the ground. Then direct the sights to 



238 



ELEMENTS OF SUKVETIXG. 



[book VI. 




Fig. 128. 



all the objects sighted to, from the other station, and mark 
the lines 1, 2, 3, 4, &c., from the base-line, as before. The 
intersections of the corresponding lines 1,1, 2,2, 3,3, 4,4, &c., 
determine, on the paper, the positions of the several objects, 
and a reference of these lines to the scale of equal parts, 
determines the true distances. 

270. Let it be required, for exam- 
ple, to determine, by means of the 
plane-table, the relative positions of 
several houses. 

From station A, and on one of the 
lines of the triangulation, as AB, 
measure the base-line AJV, which we will suppose equal to 
300 yards. Place the plane-table at A, and sight to the corners 
of the houses, and mark the lines 1, 2, 3, 4, &c. Then remove 
the table to iV, and sight to the same corners as before, and draw 
the lines as in the figure. The points at which they intersect 
the corresponding lines, before drawn, determine the corners of 
the houses. The front lines of the houses may then be drawn on 
the paper. Draw lines at right angles to the front lines, and 
on them lay off the depths of the houses, with the same scale 
as that used for the base line. 

To find the length of any line drawn on the paper, as the 
line 1, drawn through A, for example, place the dividers at A 
and extend them to the other extremity of the line, and theq 
apply the line to the scale. The 
length of the line 1 is equal to 
198 yards. 

271. In this example, we de- 
termine from the base-line CD, 
the positions of the points F, E, 
and H. 




p 

SEC. II.] FILLING UP THE SURVEY. 239 

Changing the Paper. 

272. When one paper is filled, and there is yet more work 
to be done, let the paper be removed, and a second paper put 
on the table ; after which, the table may be used as before. 

Now, in order that the .two papers may be put together and 
form one entire plan, it is necessary that two points, determined 
on the first paper, be also determined on the second ; and then, 
by placing the lines joining these points, one on the other, all 
the lines on the two papers will have the same relative position 
as the corresponding lines on the ground ; and the same for as 
many papers as it may be necessary to use. If different scales 
are used, the corresponding points will not join, and then the 
work must be reduced to the same scale, before the papers can 
be put together. 

In the first example, the position of the point F was deter- 
mined, in order to unite the first paper with the second. 

In the second example, we sighted from C and D, the ex- 
tremities of the base-line, to the points N and F, determined 
on the first paper ; we thus determined the line NF on the 
second paper. Placing the line NF of the one paper on NF of 
, the other, we have the following plan : 




PiQ. 130. 



In this plan, all the points and lines are accurately laid down. 
Any number of papers may be joined in a similar manner. 



240 ELEMENTS OF SURVEYING. [BOOK VI. 

273. I'or further description of the plane-table and its use, 
see ^^ A Treatise on the Plane-Table and its use in Topographical 
Surveying/' by E. Hergesheimer, U. S. Coast and Geodetic Sur- 
Yej, Report for 1880, Appendix No. 13. 



SECTION III. 

PLOTTING THE T R I A N G U L AT I N . 

The sides of the triangles having been completed, the work 
may then be plotted, as already explained, either by means 
of the circular protractor, or by the method of chords. 

The Circular Protractor. 

274. This instriiment consists of a brass circular limb, 
Fig. 131, of about six inches in diameter, with a movable index 
ABf having a vernier at one extremity A, and a milled screw 
at the other extremity B, with a concealed cog-wheel that works 
with the cogs of the limb, and thus moves the index AB about 
the centre of the protractor. At the centre of the protractor ^ 
is a small circular glass plate, on which two lines are cut ; the 
point of their intersection is the exact centre of the instru- 
ment. The limb is generally divided to half- degrees ; the de- 
grees are numbered from to 360. 

At the point, and at the opposite extremities of the 
diameter passing through that point, are small lines on the 
inner edge of the Hmb ; the two extremities of the diameter, 
perpendicular to this latter, are designated in the same way. 

Two angular pieces of brass, each having a small and 
sharp steel pin at its extremity, are fastened to the index, and 
revolve freely around the lines ab and cd. The smaU screws, 



SEC. III.] 



PLOTTIKa THE TRIAJ^GULATIOI^. 



24: 




Pig. 131. 



a, 5, c, and d, move them in the directions of the lines ab, cd, 
fcr the purpose of bringing the steel pins exactly into the line 
which passes through the of the index and the centre of 
the protractor. 

To adjust them to their places, place the centre of the pro- 
tractor oyer a marked point, and the of the index to the 
of the limb. Then mark the place of the index by the pins; 
after which, turn the index 180°, and see if the pins will mark 



242 ELEMENTS OF SURYEYIl^O. [BOOK VI. 

the same points as before. If they do, the index is adjusted ; 
if they do not, correct the error with the screws «, h, c, and d. 

275. To lay off an Angle with the Protractor. — Let its 
centre be placed over the angular point, and the diameter pass- 
ing through and 180'', on the given line. Turn the screw 
that works the index, until the of the vernier coincides with 
the division corresponding to the given angle; then let the 
angular brass pieces be turned down ; the points dotted by the 
steel pins will show the direction of the required line. 

If this line does not pass through the angular point, the 
pins are out of place, and must be re-adjusted. 

276. First Method of Plotting. — Suppose it were required 
to make the plot of the harbor, Fig. 122, on a scale of 450 yards 
to an inch. 

Divide the length of the base-line AB, which is equal to 
1140 yards, by 450, and the quotient 2.53 will express the 
length which is to represent the base-line on the paper. 

Draw an indefinite line AB, to represent the base; and 
having chosen any point, as A, for the first station, lay off 2.53 
inches to B. The other extremity of the base-line wiU thus be 
determined. 

Then, place the circular protractor at A, and lay off the 
angle BAB, and then the angle BAG. Next, place the pro- 
tractor at B, and lay off the angles ABB and BBC. The 
intersection of the lines AB and BB will determine the station 
B. Let the protractor be then placed at this point, and all 
the angles of station B laid down. 

The point G. where BG intersects AG, and the point C, 
where BC intersects BC, will then be found. 

By placing the protractor at C and G, we can determine the 
points D and F, when th.e place, on the paper, of all the stations 
will be known. 



I 



SEC. III.] PLOTTING THE TRIANGULATIOl^r. 243 

To unite the work done with the compass, spread the com- 
pass-notes before you, and draw through A a line to represent 
the meridian. The course AE lies to the west of this meridian, 
and makes an angle of 12° with it. 

Then, lay off from the scale the distances Aa, Ah, Aq, Ac, 
Ad, Ae, and at the several points erect perpendiculars to AE. 
Lay off, on these perpendiculars, the lengths of the offsets, and 
the curve traced through the points so determined, will be the 
margin of the lake. 

At E, draw a parallel to the meridian through A, and lay 
down the course EH, which is easterly, and makes an angle 
of 50° with the meridian. Then, lay down the several distances 
to the offsets, and draw the offsets and lay off their lengths. 
Do the same for the course HI, and all the compass-work will 
be plotted. 

The work done with the plane-table is united to the work 
done with the transit, by simply reducing it to the same 
scale, and then placing the line ^iV^ on the paper of the 
plane-table, upon the line AN, drawn on the plot of the 
triangnlation. 

277. Second Method of Plotting.— Place the centre of the 
protractor near the centre of the paper, and draw a line through 
the points and 180°. This line will have the same position with 
the circular protractor that the base-line AB had with the limb of 
the transit. 

Then lay off, from the point, an arc equal to the direction 
from A to E, also an arc equal to the direction A G, and through 
the centre point, and the points so determined, draw lines. 
Lay off in succession, in a similar manner, the directions taken 
at all the stations ; and through the centre point, and the points 
so determined, draw lines, and designate each by the letters of 
the direction to which it corresponds. 



344 ELEMENTS OF SURVEYING. [BOOK VI. 

Now, since all the lines drawn on the paper Imve the same 
position with the circular protractor, as the corresponding lines 
on the ground have with the limb of tlie transit, it follows 
that each direction will be parallel to its corresponding line upon 
the ground. 

Hence, an}' line may be drawn parallel to that passing through 
and 180°, to represent the base-line AB. Having drawn such 
a line, and marked a point for the station A, lay off the length 
of the base, and the extremity will be the station B. 

Through A and B, so determined, draw parallels respectively 
to the lines corresponding to the directions AB and BB, and 
the point of intersection will determine station B, Through 
B and B, draw parallels to the lines which correspond to the 
directions BO, CB, and their point of intersection will determine 
station C. Through and B, draw lines parallel to the lines 
corresponding to the directions CB and BD, and the point of 
intersection will determine D. In a similar manner we may 
determine the stations F and G. 

Method of Chords. 

278. The chord of a given arc is equal to the sine of half 
the arc with double the radius. 

For, let DAF be any given angle. 
And AH a hue bisecting it. Let DKC 
be the chord of the arc DC, described 
with a given radius, and HF parallel 
to CD, the sine of half the given 
angle, to a radius AF = "2 AC. 

Since AF='2AC, we have, from similar triangles, HF = 
2KCj but DC=2Ka hence HF = CD. 

279. To lay off an Angle.— To avoid, as far as possible, the 




SEC. III.] 



PLOTTING THE TRIANGULA TlOi?". 



245 




use of fractions, let us suppose the radius of the table of natural 
sines to be 1 ten, or 10 inches. 

Take, from a scale, 5 equal parts, 
with which, as a radius, from the 
centre A, describe an arc CD. Take 
from the table the natural sine of 
half the arc, and remove the decimal 
point one place to the right; the result will express the sine of 
half the arc to the radius 10, or the chord of the arc to the 
radius 5. From the same scale, take this sine in the dividers, 
and from C, as a centre, describe an arc cutting (W in D ; draw 
AD, and CAD will be the angle required. 

This is the most accurate of all the methods of laying off an 
angle, and it may also be applied advantageously to the second 
method of plotting, thus : 

Draw a fine straight line, generally in 
the direction of the meridian or of the 
base-line of the survey; and also a line 
perpendicular to it. From the point of 
intersection, as a centre, with a radius of 
5 equal parts of the scale, describe the cir- 
cumference of a circle cutting the straight lines in the points 
marked and 90°. 

To lay off an angle, as, for instance, the angle 14° 29'. The 
half of it is T 14' 30", the natural sine of which is 0.12605, 
or 1.26 to the radius of 10 inches. Set off from to h, as in 
the figure, this distance taken from the scale, and through the 
two points h, h, thus determined, draw a straight line. This 
line should pass through the centre, and will make wdth the 
line (0, 0) the angle 14° 29' ; and any line on the paper drawn 
parallel to it, will make with the line (0, 0) the same angle. 
The further application is obvious. 




BOOK VII. 



LEVELING. 



SECTION I. 

DEFINITIONS AND PRINCIPLES. 

280. Leveling is the art of determining the relative dis- 
tances of points from the centre of the earth. 

281. A line whose points are all equally distant from the 
centre of the earth, is called a line of true level ; and a surface, 
all whose points are equally distant from the centre of the earth, 
as the surface of still water, is called a level surface. 

282. One point is said to be above another, when it is farther 
from the centre of the earth ; and this difference of distance 
from the centre, is called the difference of level between the two 
points. - 

283. A straight line drawn tangent 
to a line of true level, at any point, is 
a horizontal line, and is called the hne 
of apparent level. Thus, Fig. 135, if C 
is the centre of the earth, and ^^F a line 
of true level, ABD is the line of apparent 
level. This is the line of level determined 
by an instrument. The difference be- 
tween the apparent and true level of the points A and E, is BE, 
the excess of the secant CB, of the arc AE, over the radius CE, 




Fm. 135. 



SEC. I.] DEFINITIONS. 247 

284. To find a general formula for computing this excess, we 
have (Geom., B. IV, Prop. XXX), 

AB' = BE{BE + 2JEC); 

but, since the arc AU is very small in comparison with the 
radius of the earth, the arc AE will not differ sensibly from 
the tangent AB ; the diameter 2BG may, for the same reason. 
be taken for the secant {BF-}-2BC); hence, 



AB" == BBx2BC, or, dividing by 2BC, 



^^=2W • • • • w- 

K we take the mean diameter of the earth to be 7919 miles, 

formula (1) gives 

'AB^ 
^^~7919 • • • • (^)^ ^®^^®' 

The departure of the apparent from the true level, starting 
from a given point, is equal to the square of the distance to the 
second point, divided hy the diameter of the earth. 

If, in formula (2), we give to AE, in succession, every value 
from 1 chain to any given number of chains (say 100), and 
reduce, at the same time, both terms of the fraction to inches, 
a table may be computed as on next page. 

Observe, that when the distance ^^ = 80 chains = 1 mile, 
that BE is = 8.001 inches, or two-thirds of a foot, very nearly ; 
and for any other distance, d, in miles, we have, 

12 : 6?2 :: | of a foot : |^2. 

hence, we have the following easy rule for finding the correc- 
tion of curvature in feet : 

The correction for curvature, iii feet, is equal to tiuo-thirds 
of the square of the distayice in miles. 



248 



ELEMENTS OF SURVEYING. 



[book VII. 



Table showing the differences, in inches, between the true and 
apparent level, for distances between 1 and 100 chains. 



Chains. 


Inches. 


Chains. 


Inches. 


Chains. 


Inches. 


Chains. 


Inches. 


1 


.001 


26 


.845 


51 


3.255 


76 


7.221 


2 


.005 


27 


.911 


52 


3.380 


77 


7.412 


3 


.011 


28 


.981 


53 


3.511 


78 


7.605 


4 


.020 


29 


1.051 


54 


3.645 


79 


7.802 


5 


.031 


30 


1.125 


55 


3.781 


80 


8.001 


6 


.045 


31 


1.201 


56 


3.925 


81 


8.202 


7 


.061 


32 


1.280 


57 


4.061 


82 


8.406 


8 


.080 


33 


1.360 


58 


4.205 


83 


8.612 


9 


.101 


34 


1.446 


59 


4.351 


84 


8.832 


10 


.125 


35 


1.531 


60 


4.500 


85 


9.042 


11 


.151 


36 


1.620 


61 


4.654 


86 


9.246 


12 


.180 


37 


1.711 


62 


4.805 


87 


9.462 


13 


.211 


38 


1.805 


63 


4.968 


88 


9.681 


14 


.245 


39 


1.901 


64 


5.120 


89 


9.902 


15 


.281 


40 


2.003 


65 


5.281 


90 


10.126 


16 


.320 


41 


2.101 


66 


5.443 


91 


10.351 


17 


.361 


42 


2.208 


67 


5.612 


92 


10.587 


18 


.405 


43 


2.311 


68 


5.787 


93 


10.812 


19 


.451 


44 


2.240 


69 


5.955 


94 


11.046 


20 


.500 


45 


2.531 


70 


6.125 


95 


11.233 


21 


.552 


46 


2.646 


71 


6.302 


96 


11.521 


22 


.605 


47 


2.761 


72 


6.480 


97 


11.763 


23 


.661 


48 


2.880 


73 


6.662 


98 


12.017 


24 


.720 


49 


3.004 


74 


6.846 


99 


12.246 


25 


.781 


50 


3.125 


75 


7.032 


100 


12.502 



SEC. II.] 



H^STEUMEi^-TS. 



249 



SECTION II. 



I NSTRU M ENTS. 



285. The Y Level. — A Level is an instrument used to 
indicate a horizontal line, and also, to determine the difference 
of level of any two points on the surface of the earth. 




to 

£ 

o 

W 






250 ELEMENTS OF SUKVEYING. [BOOK VII. 

The part of the instrument shown in Fig. 136, rests on a 
tripod, to which it is made fast, and on which it is leveled by 
means of two leveling plates, such as are described in the account 
of the Transit, Art. 177. 

The telescope rests on vertical supports called, from their 
shape, Y's or Wyes, and is confined in the wyes by loops or 
clips, which are fastened by tapering-pins. 

The telescope has at each end a ring of bell-metal, turned 
very truly and both rings of exactly the same diameter ; by these 
it revolves in the wyes, or can be at pleasure clamped in any posi- 
tion when the clips of the wyes are brought down upon the rings, 
by pushing in the tapering-pins. It has a rack and pinion 
movement to both object-glass and eye-piece. 

A spirit-level or ground bubble-tube is attached to the under 
side of the telescope, and furnished at the different ends with 
screws for movement in both horizontal and vertical directions. 

The level scale which extends over the whole length is gradu- 
ated to tenths of an inch, and figured at every fifth division, 
counting from zero at the centre of the aperture of the tube through 
which the glass vial appears ; the scale is set close to the glass. 

The wyes are perpendicular to a level-bar, to which they are 
screwed fast ; each wye has two nuts, both adjustable with the 
ordinary steel pin. Connected with the level-bar is the head of 
the tripod-socket. 

The tripod-socket is compound ; the interior spindle D, 
Fig. 137, upon which the whole instrument is supported, is 
made of steel, and nicely ground, so as to turn evenly and 
firmly in a hollow cylinder of bell-metal; this again has its 
exterior surface fitted and ground to the main socket EE of the 
tripod-head. 

The bronze cylinder is held upon the spindle by a washer and 
screw, the head of the last having a hole in its centre, through 
which the string of the plumb-bob is passed. 



SEC. II.] 



INSTRUMENTS. 



251 



The upper part of the instrument, with the socket, may thus 
be detached from the tripod-head; and this also can be un- 




&A 



screwed from the legs, so that both may be conveniently 
packed in the box. 

A little under the upper parallel plate of the tripod-head, and 
in the main socket, is a screw which can be moved into a corres 



252 ELEMENTS OF SURyEYi:5sG. [BOOK Til. 

ponding groove, turned on the outside of the hollow cylinder, 
and thus made to hold the instrument in the tripod when it is 
carried upon the shoulders. 

Before using the Level, it must be adjusted. The adjust- 
ment consists in bringing the different parts to their proper 
places. 

The line of collimation is the axis of the telescope. With 
this axis, the line drawn through the centre of the eye-glass 
and the intersection of the cross-wires, within the barrel of 
the telescope, ought to coincide. 

FIRST ADJUSTMENT. 

To fix the intersection of the cross-wires in the axis of the 

telescope. 

286. Having screwed the tripod to the instrument, extend 
the legs and place them firmly. Then loosen the clamp-screw 
and direct the telescope to a small, well-defined, and distant 
object. Then slide the eye-glass till the cross-wires are seen 
distinctly ; after which adjust the object-glass to its proper 
focus, when the object and the cross- wires will be distinctly 
seen. Xote now the precise point covered by the intersection 
of the cross-wires. 

Having done this, revolye the telescope in the Y's half 
round, when the attached level will come to the upper side. 
See if, in this position, the horizontal wire appears above or 
below the point, and in either case, loosen the one and tighten 
the other of the two screws which work the horizontal wire, 
until it has been carried orer half the space between its last 
position and the observed point; bring it the rest of the way 
by the leveling screws. Carry the telescope back to its place ; 
direct again the intersection of the cross-wires to the point, and 
repeat the operation till the horizontal wire neither ascends 



SEC. II.] Iiq-STKUMENTS. 253 

nor descends while the telescope is revolved. A similar process 
will arrange the vertical wire, and the line of collimation is then 
adjusted. 

SECOND ADJUSTMENT. 

To make the axis of the attached level parallel to the line of 

collimation. 

287. Turn the leveling-screws until the bubble of the leyel 
stands at the middle of the tube. Then open the loops and 
reverse the telescope. If the bubble still stands at the middle 
of the tube, the axis of the level is horizontal ; but if not, it is 
inclined, the bubble being at the elevated end. In such case, 
raise the depressed or depress the elevated end by means of the 
small screw provided for the purpose, half the inclination ; and 
then with the leveling screws bring the level to a horizontal 
position. Eeverse the telescope in the Y's and make similar 
corrections again ; and proceed thus until the bubble stands in 
the middle of the tube, in both positions of the telescope ; the 
axis of the Jevel is then horizontal. 

Let the telescope be now revolved in the Y's. If the bubble 
continues in the middle of the tube, the axis of the level is not 
only horizontal, but also parallel to the line of collimation. If, 
however, the bubble recedes from the centre, the axis of the 
level is inclined to the line of collimation, and must be made 
parallel to it by means of two small screws, which work hori- 
zontally. By loosening one of them and tightening the other, 
the level is soon brought parallel to the line of collimation ; 
and then, if the telescope be revolved in the Y's, the bubble will 
continue at the middle of the point of the tube. It is, however, 
difficult to make the first part of this adjustment, while the axis 
of the level is considerably inclined to the line of collimation ; 
for, even if the level be truly horizontal in one position of the 
telescope, after it is reversed there will be but one corresponding 



ii 



254 ELEMEi^TS OF SUKYEYIIs'G. [BOOK VIL 

position in which the bubble will stand at the middle of the 
tube. This suggests the necessity of making the first part of 
the adjustment with tolerable accuracy ; then, having made the 
second with care, re-examine tlie first, and proceed thus till the 
adjustment is completed. 



THIRD ADJUSTMENT. 

To make the level and the line of coUimation perpendicular to 
the axis of the instrument, or parallel to the level-bar. 

288. Loosen the clamp-screw and turn the bar until the 
level comes directly over two of the leveling screws. By means 
of these screws, make the level truly horizontal. Then, turn 
the level 180° upon its vertical axis ; if, during the revolution, it 
continue horizontal, it must be at right angles to the axis of 
the instrument about which it has been revolved. But if, after 
the revolution, the level be not horizontal, rectify half the 
error with the screws at M and R, Fig. 137, and half with the 
leveling screws. Then place the bar over the other two leveling 
screws, and make the same examinations and corrections as 
before ; and proceed thus, until the level can be turned entirely 
around without displacing the bubble at the centre. When this 
can be done, it is obvious that the level and the line of 
collimation are at right angles to the axis of the instrument 
about which they revolve ; and since the axis is carefully ad- 
justed by the maker, at right angles to the bar, it follows 
that the line of collimation, the level, and the bar, are parallel 
to each other. 

It is always necessary to examine the adjustments frequently, 
in order to secure satisfactory resr.lts. 



SEC. II.] 



INSTRUMENTS. 



255 



LEVELING RODS. 

289. The leveling rods are used to determine the points at 
which a given horizontal line intersects lines 

that are perpendicular to the surface of the 
earth, and to show the distances of such points 
of intersection from the ground. 

There are three kinds of rods used by En- 
gineers, known as the New York, Philadelphia, 
and Boston or Yankee rods. The Philadelphia 
Rod is divided to tenths, and reads to two-hun- 
dredths of a foot. The New York and Boston 
rods are divided to hundredths of a foot, and 
read by verniers to thousandths. They are all 
sliding rods. 

290. New York Rod.— This rod, which is 
shown in Fig. 138, is cut in two parts so that 
both ends may be exhibited. It is made of 
maple or satin-wood, in two pieces, sliding one 
from the other, always in the same direction, so 
that the same end is always held on the ground, 
and the graduations start from that point. 

The graduations are made to tenths and hun- 
dredths of a foot, the tenth figures being black, 
and the feet marked with a large red figure. 

A target is used to indicate where the hori- 
zontal line cuts the rod. 

The face of the target is divided into quad- 
rants, by a horizontal and a vertical diameter; 
and these diameters are the boundaries of alter- 
nate colors with which the diagonal quadrants 
are painted. 

The opening in the face of the target is a fig. i38. 





11 



256 



ELEMI5NTS OF SURVEYING. 



[book VII. 



little more tliaii a tentli of ji foot long, so Uuit in any position a 
tenth, or a foot figure, ciin be seen on the surface of the rod. 

The right edge ol' the opening is chamfered, and divided into 
ten equal spaces, corresponding with nine-huiidredths on the 
rod ; the divisions start from the horizontal line whicli separates 
the colors of the face. The vernier reads to thousandths of a 
foot. 

For heights less than six and a hall* feet, the target is moved 
along the sliding part, to which it is slightly attached by springs, 
and to which it may be permanently attached by a clamp-screw, 
and the reading is made by the vernier on the target. 

When a greater height is required, the horizontal line of the 
target is fixed at that point, and (he upper half of the rod, car- 
rying the target, is moved out of the lower, the reading being 
now obtained by a vernier on the graduated side, up to an eleva- 
tion of twelve feet. 

TESTS OF ADJUSTMENT. 

291. There is a method of testing the adjustments of the 
Y level, which ought not to be neglected, since all the results 
depend on the accuracy of the instrument. The method is 
this: 



F 



B 




' "^wi;:,'.'."!!'.'.'.".]!]] 



is 



PiQ. 189. 



E 



The level being adjusted, place it at any convenient point, 
as G (Fig. 139). At equal distances of about 300 feet in opposite 



8EC. IT.] IKSTEUMENTS. 257 

directions from thie instrument, drive two pegs firmly into the 
ground and take readings of the rod upon them. The difference 
of the readings will he the difference of level of the tops of 
the pegs, even though the level be out of adjustment. Now set 
the level at about 50 feet beyond either f><^g^ nearly in line with 
them, and again take rod-readings upon them ; the difference of 
these new readings, corrected for curvature of the earth, should 
equal the difference of level of the pegs as found before. If the 
readings are not what they should U^, the adjustment may ?je 
perfected thus: Call the further peg «, the nearer peg h, 




and the position of the instrument c, (Fig. 140). Let R be the 
first reading at «, and W that at 6 ; let r be the second reading 
at a, corrected for curvature, and r' the corresponding reading 
at b, also corrected ; let 

(R _ R') -(r-r'}= ±D; 

then we have 

ab (= GOO ft.) : ac (= 650 ft.) : : ± D : ± d. 

Add the correction d to r, and having set the target to this 
reading on a, bring the horizontal wire to coincide with it by 
the afl justing screws. 8ee also that the bubble is in the middle 
of the run at the same time. 



Ifc 



258 ELEME^'TS or SUBTEYIXG. fBOOK Vn. 

SECTION III. 

LEVE LING IN THE FIE LD. 

292. The operatious of leveling may be undertaken . 

1st. For the purpose of determining the difference of level 
between two given points ; 

2d. For the purpose of obtaining a section or profile along 
a given line, as in the preliminary surveys for railroads and 
canals ; 

And, as will be described hereafter, 

3d. For the purpose of determining the contonr lines in a 
topographical survey ; 

4:thly. For the purpose of determining the volume of any 
given mass of earthwork or masonry ; as the measurement 
of excavations and embankments for canals and railroads ; 
and, 

5thly. For the purpose of determining and indicating 
boundaries for filling and excavation ; such as setting slope 
stakes, &c. 

Difference of Level between Two Points. 

293. TVhen it is proposed to find the difference of level of 
any two objects, or stations, all levels made in the direction of 
the station at which the work is begun, are called, for the sake 
of distinction merely, hach-sigJits ; and levels taken in the direc- 
tion of the other station, foresights. 

Before going on the field with the level, rule three columns, 
as below, and head them, stations, back-sights, fore-sights. 



SEC. III.] 



LEVELING IN THE FIELD. 



259 



FIELD NOTES. 



Stations. 


+ Back-Sights. 


— Fore-Sight3. 


1 


10 


3 


2 


11.6 





3 


6.8 


4.9 


4 


3.9 


8.3 


Sums 


. . . 32.3 
16.2 


16.2 


Dif. of leve 


I . . 16.1 





Find the difference of level between any two points, as A and 

G (Fig. 141). 

294. The level being adjusted, place it at any point, as B, as 
nearly in the line joining A and G, as may be convenient. 

h 




Place a leveling rod at A. Make the level horizontal by means 
of the leveling screws; turn the telescope to the rod at J, and 
direct the rodman to raise the target until the horizontal line 



360 ELEMENTS OF SUKVEYII^G. [BOOK VII. 

ah pierces its centre ; then note the distance Ah (equal to 10 
feet in the present example) and enter it in the column of 
back-sights opposite station 1. 

Send the rodman forward to some point, as N, in the pro- 
posed direction, and sight to the rod as before; enter the dis- 
tance Na, equal to 3 feet, in the column of fore-sights opposite 
station 1 {B). Then remove the level to a convenient point, 
as C (2). Direct the rodman to run up the vane to the proper 
height; then make the back-sight, and enter it, Nd = 11.6 feet, 
in the column of back-sights, opposite station 2. Let the rod- 
man then be sent forward to a convenient point, as if, and 
make the fore-sight to /; and enter Mf = 0, in the column of 
fore-sights, opposite station 2 (C). Remove the level, in succes- 
sion, to D and U, and make similar levels at those points, and 
enter the results in the column of back-sights and fore-sights, 
opposite station 3 (D), and 4 (U). 

It is evident from the figure, that the difference of level JVF, 
between A and iV, is equal to the back-sight hA, diminished by 
the fore-sight aN; also that the difference of level between 
iV and if, is equal to the back-sight dJV, diminished by the 
-fore-sight 0, and since each set of observations is entirely inde- 
pendent of every other set, we may infer that the difference of 
level hehveen two consecutive points, as determined hy the same 
position of the level, is equal to the hack-sight, diminished hy the 
fore-sight. If the fore-sight is greater than the back-sight, the 
difference will be affected with a minus sign, a result which 
shows that the second point is lower than the first ; and. 

Generally, the difference of level hetiveen any two points, deter- 
mined as ahove, is equal to the sum of the hack-sights diminished 
hy the sum of the fore-sights. If the result is plus, the second 
point is higher than the first ; if negative, it is lower. 

In the example given, the difference of level between A and 
6^, is 16 feet and 1 tenth. 



SEC. III.] 



LEVELIXG IN" THE FIELD. 



261 



295. In the above example, we did not regard the difference 
between the true and apparent level. If it is necessary to 
ascertain the result with extreme accuracy, this difference must 
be considered ; and then, the horizontal distances between the 
level, at each of its positions, and the rods, must be measured, 
and the apparent levels diminished by the differences of level ; 
which differences can be found from the table. 











EXAM PLE. 








Stat. 
1 


Back-sts. 


Distances. 


Fore-st. 


Distances. 


Cor. back-sts. 


Cor. for-sts. 


9.8 


20 ch. 


1.6 


32 ch. 


9.7583 


1.4933 




2 


8.7 


25 ch. 


2.4 


28 ch. 


8.6349 


2.3183 




3 


5.2 


18 ch. 


3.1 


16 ch. 


5.1663 


3.0734 




4 


10.3 


29 ch. 


1.9 


87 ch. 


10.2124 


1.1115 




5 


11.0 


45 ch. 


2.5 


72 ch. 


10.7891 


1.9600 




44.5610 


8.9565 



In this example, the first column shows the stations ; the 
second, the back-sights ; the third, the distances from the level 
in each of its positions to the back rod ; the fourth, the fore- 
sights ; the fifth, the distances from the level to the forward rod ; 
the sixth and seventh, are the columns of back and fore-sights, 
corrected by the difference of level. The corrections are thus 
made : The difference of level in the table corresponding to 20 
chains, is 5 tenths of an inch, or .0417 of a foot nearly; which 
being subtracted from 9.8 feet, leaves 9.7583 feet for the corrected 
back-sights; this is entered opposite station 1 in the sixth 
column. The difference of level corresponding to 32 chains, is 
1.280 inches, or .1067 feet, nearly ; which being subtracted from 
the apparent level, 1 foot 6 tenths, leaves 1.4933 feet, for the true 
fore-sight from station 1. The other corrections are made in 
the same manner. 



li 



263 ELEMENTS OF SURVEYING. [bOOK VIL 

The sum of the back-sights being 44.5610 feet, and the 
sum of the fore-sights 8.9565 feet, it follows that the difference, 
35.6045 feet, is the true difference of level. 

296. In finding the true from the apparent level, we have 
not regarded the effect caused by refraction on the apparent 
elevation of objects, as well because the refraction is different 
in different states of the atmosphere, as because the corrections 
are for short distances inconsiderable. The error occasioned by 
refraction is opposite to, and tends to diminish, the error occa- 
sioned by the curvature of the earth. If desired, it may be 
corrected by diminishing the effect of the earth's curvature by 
one-seventh of itself. 

297. The small errors that would arise in regarding the 
apparent as the true level, may be avoided iy placing the leveling 
rods at equal distances from the level. In such case, it is plain, 
1st, that equal corrections must be made in the fore and back- 
sights; and, 2dly, that when the fore and back-sights are 
diminished equally, the result, which is always the difference of 
their sums, will not be affected. 

This method should always be followed, if practicable, as it 
avoids the trouble of making corrections for the difference of true 
and apparent level. 

The differences between the true and apparent level, being 
very inconsiderable for short distances, if only ordinary accuracy 
is required, it will be unnecessary to make measurements at all, 
Care, however, ought to be taken, in placing the levehng rods, 
to have them as nearly equidistant from the level, which can be 
effected by the rodman, by pacing the distance from back-sight 
rod to level, and from level to fore-sight rod ; and if the distances 
are unequal, let the next distances also be made unequal ; that is, 
if the back-sight is the longer in the first case, let it be made 
proportionably shorter in the second, and the reverse. 



SEC. IV.] SECTION LEVELING. 263 



SECTION IV. 

SECTION LEVELING. 

298. In the surveys which precede the construction of roads, 
raih'oads, canals, dikes, or other similar earthworks, the surveyor 
must make such measurements as are necessary to enable him 
to estimate the volume of the material to be removed. In 
addition, therefore, to the horizontal measurements made in 
connection with the location of the work, vertical dimensions, 
or heights, are also necessary, and are taken at every important 
change in the inclination of the surface along the line of the 
survey. 

These heights are taken by the level and rod, and are 
simply vertical distances of points along the surface above an 
assumed level line called the datum line. 

299. In the survey of a long line of railway or canal, one 
of whose termini is in the vicinity of tide-water, the datum line 
is usually assumed at the level of mean high -water. In cases 
of surveys entirely inland, the datum line is taken at some 
convenient depth below the beginning point of the survey, and 
at such a distance that it shall be below the entire line on the 
surface. For such surveys, the system of notes described in 
the preceding section is insufficient. 

300. As the survey progresses, fixed points of reference, 
called tenches, are located in the vicinity of the line. Permanent 
objects are usually selected for benches ; such as rocks, build- 
ings, or trees, and at such distances from the line of the work 
as to be undisturbed by the subsequent construction. 



II 



264 ELEMENTS OF SURVETIKG. [BOOK TIL 

301. Temporary benches, employed merely while changing 
the position of the leveling instrument, are called turning points. 
In either case, a well-defined point must be provided — one not 
easily disturbed by a blow, and, moreover, one upon which the 
rod can be held vertically. 

302. In order to understand the field operations and the 
mode of keeping notes, it will be necessary to comprehend the 
principle involved in all leveling practice. 

Suppose that the depths of different points of the bottom of a 
shallow lake are required ; these could be readily obtained by 
measuring the distances from the surface of the water to the 
bottom by means of a sounding line, or in winter by cutting 
holes in the ice and measuring the distances to the bottom with 
a pole. 

In this illustration the various points on the bottom are 
located with reference to the plane of the surface of the 
water. 

The practice of leveling is identical in principle. The hori- 
zontal surface of reference, from which to deduce relative heights, 
is generated by the line of collimation as the telescope is revolved 
about the vertical axis, and the '^soundings" to points below 
this plane of reference are made with a leveling rod, whose 
lower end rests upon the point to be located, and whose target 
is moved so that the plane of reference shall cut its horizontal 
line. 

When the instrument is set up at a new station, the distance 
of its new plane of reference above or below the previous 
one, must be determined in order to secure continuity of the 
work. 

303. The following example will exhibit the method of re- 
cording the notes of a section level. The datum line is assumed 
to be thirty feet below the first bench. When the field-book 



BEC. IV.J 



SECTION" LEVELING. 



265 



is Qf the ordinary pocket size, the seven columns of notes will 
generally occupy two opposite pages ; the first five being upon 
the left-hand page. 



2\.Gyitt dboreDahnn. 



^ench 




Dist. 


+ Sight. 


Ht. of Ins. 


- Sight. 


Surface 
Height. 


Grade 
Height, 


Remarks. 


Bench. 

1 


1.637 


31.637 

1 / ' 


2.1 

1.8 


30. 

29.5 

29.8 




Bench on top 

of fence-post 

30 ft. north of 

stake. 


160 






0.9 


30.7 






2 






3.4 


28.2 






3 






10.8 


20.8 






T. P. 


1.910 


22.134 


11.413 


20.224 




! 


4 






5.8 


16.3 






5 






9.0 


13.1 




, 


5S0 






10.4 


11.7 






6 






9.8 


12.3 






7 






10.6 


11.5 







The bench haying been selected and marked, its location is 
described in the column of remarks. 

The level is adjusted in some convenient place in the vicinity, 



266 ELEMENTS OF SURVEYING. [BOOK VIL 

and the reading of the rod is taken upon the bench. In the 
above example it is 1.637. As the bench is 30 feet above the 
assumed datum line, the height of the instrument (or line of 
collimation) above this datum line is 31.637 feet. 

The reading is recorded against Bench, in the column of 
+ sights, and the ''height of instrument" is recorded in its 
proper column, in the same line. 

By referring to the above diagram it will be readily seen, 
that to obtain the height of the different points 0, 1, \^, &c., 
above the datum line, it is only necessary to take the readings 
of the rod, at these stations, and subtract them from 31.637. 
Such readings, therefore, are appropriately termed ininus sights, 
and are recorded in the 4th column. As these readings are 
taken only to the nearest tenth of a foot, they are taken much 
more rapidly than the bench readings. The subtractions by 
which the surface heights are found, may be worked in the 
field or not, as the surveyor chooses. The unit of measurement, 
in the column of distances, is usually the engineer's chain of 
100 feet. Headings are taken at intermediate points (as at 160 
feet in the above example) when there are abrupt changes in 
the inclination of the surface. 

306. When it becomes necessary to change the position of 
the level, such measures must be taken as will insure the exact 
"height of instrument," in the new position. 

To effect this, a carefully-selected hard point is found (not 
necessarily on the exact line of the survey, but as far forward 
as convenience and accuracy will permit), and a reading of the 
rod is taken upon it, to thousandths. 

If likely to be used for a single occasion only, it is called a 
" turning -poiyit,^^ and marked T. P. in the distance column ; 
otherwise it is called a Bench, and its location is described in 
the column of remarks. 



SEC. IV.] SECTION^ LEVELIN^G. 267 

A turning-point is taken between stations 3 and 4, in the 
above example. The reading of the rod, upon it, is 11.413. 
This is recorded in the — sight column, and the surface height 
of the point is at once found as before, and recorded in the 
column of " Heights. " 

The level is next carried forward to a new position, adjusted, 
and directed again upon the rod still held upon the turning- 
point. The reading is taken to thousandths. This, when added 
to the height of the turning-point, evidently gives the height of 
instrument in its new position. It is recorded, therefore, as a -|- 
sight. The survey is now continued by taking — sights at the 
various points along the line until it becomes again necessary to 
change the position of the level. 

In the above example, the reading of the rod upon the 
turning-point, from the second position of the level, is 1.910. 
The height of the point upon which the rod stands is 20.224. 
The sum of these, or 22.134, is the "Height of Inst." for the 
second set of — sights. The successive subtractions of the 
readings from the Height of Instrument, give the surface heights 
as before. 

The most extended section levels are but repetitions of this 
process. 

305. The rules for taking and recording field-notes in section 
leveling are as follows: 

I. The " distances " recorded in the first column arc the 
horizontal measurements, in chains, frovv the beginnin£ 
of the survey to the points whose heights a,re to he dete?^- 
mined. TJie heights are taken at each ivhole chain, and 
at such intermediate points as the irregularities of the 
surface require. 

II. The first reading of the rod, after each setting of 



268 ELEMENTS OF SURVEYING. [BOOK VII 

the level, is upon a bench or turning-point, and is a '^ -\- 
SIGHT ; '' all other readings are " — sights/' 

III. The + sight, added to the height of the point upon 
which the reading was taken, gives the " Height of Instru- 
ment." 

IV. The — sights taken, at any position of the level, 
subtracted from the "Height of Instrument" for that 
position, give the corresponding " Surface Heights." 

V. All the + SIGHT readings, and the last — sight of 
each set, being upon benches or turning-points, are taken to 
thousandths of a foot. The remaining '' — sights " are 
taken to tenths only. 

Note. — It will be observed that when the column of " surface 
heights " is complete, the second, third, and fourth columns of 
the field-notes are no longer needed. The first and fifth columns, 
which together contain the horizontal and vertical measurements 
for the line of work, afford all the data necessary for mapping 
the profile and determining the grade-line. 

306. The location of the benches should be so described in 
the column of "remarks," that any particular bench may be 
found at any time, by referring to the field-notes. The impor- 
tance of this is apparent when it is remembered that the process 
of construction destroys or removes the stakes along the line of 
the survey, and that the question of the completion of the work 
can be determined only by reference to the benches. It is 
obvious, also, that they should be established somewhat off the 
line of the survey. The distance apart, of regularly established 
benches, should be governed by the above-mentioned uses of 
them. 

Any turning-point may be profitably made a bench (when 
it can be made permanent), by carefully recording, so as to 
admit of its identification. 



SEC. IV.] SECTION LEVELIITG. 269 

In order to eliminate the effects of curvature and refraction, 
*Hurning-points," whether used as benches or not, should be 
taken at equal distances from the instrument when practicable, 
or compensation should be secured by proportional distances, as 
in Art. 297. 

307. In conducting a section level through a rocky district, 
turning-points in abundance are found at hand, and cause no 
delay in their preparation, whereas a bench in the same section, 
requires marking and locating. 

In leveling through flat and level sections of country, 
although the engineer can get ^* sights" for long distances, a 
proper regard for accuracy will induce him to limit the distance, 
between successive positions of the level, to about six hundred feet. 
Under such circumstances, each turning-point is made a bench. 

308. The methods of establishing benches are various. In a 
rocky section, some conspicuous point is marked either by 
drilling or grooving the rock. In villages or cities, stone steps, 
or projecting courses of masonry to dwellings, curb-stones, and 
fence-posts afford good benches, and admit of easy identification. 

In sections where trees abound, a notch is cut in the side 
of a trunk near the root, in such a manner as to leave a pro- 
jecting point upon which the rod may be held vertically. A nail 
driven full length into the projection, gives it the necessary firm- 
ness for a bench. 

In marshes or prairies, where there are neither rocks nor 
trees, the engineer is compelled to resort to long stakes, firmly 
driven into the ground to such a depth as to be undisturbed by 
the frost ; no portion of the stakes being allowed to project above 
the surface. The top of each is trimmed to a kind of blunt 
point, into which a nail is driven its full length. 

A re-survey of a route, to detect possible errors in leveling, 
is accomplished by taking the heights of the '* benches" only, 
and is called a " check level." 



370 ELEMENTS OF SURVEYING. [BOOK VII. 

Drawing the Profile. 

309. When the " section level " of a line of work has been 
completed, the "profile" is next to be drawn. The method of 
doing this is very simple. 

A horizontal line to represent the dattim line is first drawn, 
and the distances from the first column of notes are laid off 
along it, to a convenient scale; this for ordinary working draw- 
ings is about two hundred feet to an inch. 

The "surface heights" corresponding to these distances are 
next laid off at right angles to the datum line, and above it, 
but to a scale usually ten times as great as that employed for 
the horizontal distances; that is, an inch upon the vertical lines 
represents one-tenth as many feet as upon the datum line. A 
line joining the upper extremity of the verticals, is the profile. 

By thus employing two different scales, the irregularities 
of the surface are made more apparent to the eye, and the sub- 
sequent adjustment of the *^ grade-line" is rendered much easier 
and more accurate. 

310. Every earthwork of importance requires, in addition to 
the working profiles, a general map, in which the plan drawn from 
the transit survey is represented upon the same sheet as the profile. 

The horizontal distances of both portions of the map being 
drawn to the same scale, and one being placed directly above the 
other, corresponding points in plan and profile are readily compared. 

In published maps of this kind, representing extended works, 
and drawn for convenience to a very small scale, the vertical 
scale of the profile is frequently several hundred times as great 
as the horizontal. 

Establishment of the Grade. 

311. The determination of the height which the finished 
road or canal shall have above the datum line at different points, 
is called "Establishing the grade." 



SEC. IV. J SECTION LEYELIN-Q. 271 

The position and inclination of grade-lines are influenced by 
a variety of circumstances : 

1st. The character of the work. A street admits of an in- 
clination of five, or even eight feet in a hundred, and requires 
about one foot for its drainage, while a rise of two feet in a 
hundred upon a railroad is exceedingly rare. A canal is, of 
course, level, the change of height being effected by abrupt 
transitions at the locks. 

2d. The economy of construction. It is desirable to make 
the earth excavated, form the required embankments, or, in 
the language of the engineer, " to make the cuttings balance the 
fillings." 

It is, however, sometimes more economical to throw away, 
or "make a spoil bank" of the earth of an excavation, than 
to transport it the required distance for the embankment. 
Embankments, for similar reasons, are often constructed of 
earth obtained outside of the road limits (''borrowing pits") ; 
or, when such means are not available, are often made of timber 
framing (trestle-work). 

3d. The natural obstacles, which render the construction 
difficult ; such as rocky ledges, marshes, lakes, streams, and 
quicksands. 

In any case, the engineer determines, by inspection of 
the maps, at what points the grade-line shall intersect the 
natural surface. Thus the inclination of the grade, and, con- 
sequently, its height above the datum line, for each "distance," 
are easily found. 

Another column of notes is now made, recording these 
" Grade Heights ; " each being placed against the corresponding 
surface height. 

312. The. folio wing example, with its accompanying diagram, 
illustrates the method of establishing a grade and recording 
the notes. It will be observed that the profile, with its " dis- 



273 



ELEMEI^TS OF SURVEY I ]SrG. 



[book VII. 



tances" and "surface heights," are the same as in the preceding 
problem. 

We will suppose it is required to establish, in the following 
profile, a gi*ade-line whose inclination shall not exceed 3 in 100 ; 
the grade to begin at station 0, at the surface. 




Fio. 143. 



DiBt. 


+s. 


H. of Ins. 


-S. 


H. of Sur. 


H. of Gr. 


Cut. 


Fill. 


Rem. 











29.5 


29.5 








1 








29.8 


26.8 


3.0 






^60 








30.7 


25.2 


5.5 






2 








28.2 


24.1 


4.1 




Oat 2" 


3 








20.8 


21.4 




0.6 




T.P. 


















4 








16.3 


18.7 




2.4 




5 








13.1 


16.0 




2.9 




550 








11.7 


14.7 




3.0 




6 








12.3 


13.3 




1.0 




7 








11.5 








Oat 6" 



. It is an easy matter to represent any required inclination 
of grade on the profile map; nothing more being necessary 
than to lay off the proper distances on two different verticals. 



SEC. IV.] SECTION" LEVELING. 273 

and draw a line through the points of measurement. For in- 
stance : a grade of 3 in 100, running downward from station 0, 
would intersect the vertical at 6, eighteen feet lower, and the 
vertical at 7, twenty-one feet lower. 

Moreover, by consulting the notes, we find that a grade- 
line from 0, whose height is 29.5 feet, ending at the surface 
at 7, whose height is 11.5 feet, descends 18 feet in 700, or 
2.57 in 100. 

Either of these lines would fulfil the required conditions. 
The first would, however, require in its construction a large 
excess of excavation over the embankment (as may be seen by 
drawing a faint line in the diagram). The second would give 
an excess of embankment. 

It is best, generally, that the cutting should be slightly in 
excess, as nearly all kinds of earth shrink a little in the process 
of removal. 

The cuttings and fillings of the profile may be balanced 
with tolerable accuracy, by stretching a thread across the pro- 
file so as to intersect at the point, and then varying the in- 
clination, until the areas cut off by the profile line on opposite 
sides of the thread appear equal.* 

The column of Grade Heights must now be filled. It is 
easily and rapidly done. The height of Grade, at 0, is, by the 
conditions, 29.5. At station 1, it must be 2.7 lower, or 26.8 ; at 
1.60, 4.3 lower, or 25.2; and at 2, 5.4 lower, or 24.1, &c. 

The remaining columns of *^cut" and ''fill" contain simply 
the differences between corresponding "surface" and "grade 



* The advantage of a thread over a ruler lies in the fact, that while using the thread, 
the areas on both sides of it are seen at once. 

In the present example, a line from 0, descending 2.7 to 100, seems to accomplish the 
desired purpose. The line being drawn, the "cut" and "fill" areas are measured, to 
determine if they are properly balanced. 

The complete computation of the earthwork, by which the exact position of the grade- 
line is determined, is explained in a following section. 



274 ELEMEl^TS OF SURVEYII^G. [BOOK VII. 

heights." Where the surface is higher than the grade-line, the 
construction requires a " cutting ; " when the established grade- 
line is higher than the surface, an embankment, or '^ filling," is 
necessary. 

The notes in the final column, indicate the points where the 
grade-line intersects the natural surface. Such are called zero 
points. 

The distances are of importance in the computation of the 
earthwork. The above notes literally signify that either cut or 
fill is 0, at 2.87, also at 6.52. 

These distances are obtained with sufficient accuracy for 
ordinary purposes by a measurement of the profile map. When 
the cuttings and fillings are recorded in the proper columns, 
the notes belonging to the section-level are complete.* 

Note. — It will be observed that the first set of notes on 
page 265, did not contain the columns for cut and fill. 

The practice in keeping the notes differs with the work to 
be performed. 

In extensive railway surveys, it is convenient to rule the 
pages of the note-book as in the first example ; carrying out the 
field-notes to the extent of the surface heights, at least ; then 
transfer to another book, the "distances," "surface heights," and 
"grade heights," ruling columns for "cut," "fill," and "remarks." 

* The following calculation may be employed In the more important cases. The 
triangles formed by the verticals (cut or fill), the grade-line, and the surface-line are 
similar, and give the following proportion : 

The sum of the cut and fill, 

: the cut, 
: : the distance from cut to fill, 
: distance from the cut to point. 

Fill may be substituted for cut in the second and fourth terms. 

The application to the first zero point, in the above notes, is as follows : 

4.1+0.6 : 4.1 :: 100 : required dist., or 87. 

In the second case in the notes, the cut necessary to the calculation is wanting, but 
U easily supplied, by determining the height of grade in the usual way at station 7. 



SEC. IT.] 



SECTION LEVELING. 



275 



These transferred notes are recorded in ink, and reserved for 
use in mapping and computations. 

313. Most road or canal surveys are made on several trial- 
lines before one is finally adopted. The profile of each line is care- 
fully drawn, and the cost of construction approximately estimated. 

When the route is finally selected, and the section levjels 
satisfactorily completed, the exact width of the earthwork, bdth 
in excavations and embankments, is carefully staked out and 
the amount of material to be moved in the progress of construc- 
tion, accurately measured. 

The method of conducting this work is explained in a follow- 
ing section. 

314. Before closing the subject of section leveling, we will 
consider the profile represented in the figure, and the set of field- 



10.912 




\jDahnnX/t'ne80fee% hetaw firsl- JSen^, 



3*0 

Fig. 144. 



6 6«> 



notes appended, which are only partially completed, and whicli 
will afford some examples for practice. 



276 



ELEMENTS OF SUKVEYING. 



[book VIL 



Dist. 


+ s. 


Ht. of Ins. 


- S. 


Surface 
Heights, 


Remarks. 


Bench. 


1.032 

















3.2 






1 






3.8 






2 






5.3 






3 






8.9 






350 






10.3 






4 






9.0 - 






5 






4.8 






T. P. 


11.815 




2.346 






T. P. 


10.942 




2.318 






6 






9.7 






640 






6.4 






7 






2.1 







1. 


What 


of the level 


2. 


What 


3. 


What 


level r 


) 


4. 


What 


5. 


What 


level '. 


J 


6. 


What 


7. 


t( 


8. 


a 


9. 


a 


10. 


iC 


11. 


Write 


and 6^- 



is the " Height of Instrument " for the first position 
? Ans. 31.032. 

is the height of the first T. P. ? Ans, 28.686. 
is the "Ht. of Inst." for th^ second position of the 

A71S. 40.501. 

is the height of the second T. P. ? Ans. 38.183. 

is the '^Ht. of Inst.'' for the third position of the 

Ans. 49.125. 

is the Height of Surface at ? 



atO? 


Ans. 


27.8. 


at 3? 




22.1. 


at 350? 




20.7. 


at 5? 




26.2. 


ate? 




39.4. 


at 7? 




47.0. 



the "Surface Heights" for the distances 1, 2, 4, 



SEC. V.l 



CROSS-SECTIOi^ LEVELING. 



277 



SECTION V. 

CROSS-SECTION LEVELING. 

315. All earthworks, whether excayation or embankment, 
unless held in position by retaining walls, require to be con- 
structed with a sloping surface, the inclination of which depends 
upon the kind of earth. 

If, in a railway-cutting, for instance, the banks which bound 
it be left too nearly vertical, when first constructed, the weather- 
ing influences, to which they are subjected, soon cause the ma- 
terial to slide down, until the whole slope gradually assumes a 
much lower inclination. 

After a time, however, the tendency to roll or slide is checked 
by the friction of the particles themselves, and the slope thus 
formed will withstand the ordinary effects of sun, wind, and 
rain. The inclination thus assumed is called the " natural slope " 
of that kind of earth.* 

316. Slopes are expressed mathematically by the ratio of 



E 



F 



S' 




^^■^iiai^^HB^^'' 




Fig. U5. 



their horizontal to their vertical dimensions, and which is called 
the ratio of slope. 



* Thi? slope is determined, experimentally, by drying a portion of the earth, and 
then pouring it from a slight elevation upon a level surface. The heap thus formed 
is a rather flat cone, whose sides stand at the lowest inclination they would be liable to 
assume under the action of atmospheric influences. The angle with the horizontal plane 
will be somewhere between 25° and 45°. 



278 ELEMEi^TS OF SURVEYING. [BOOK VII. 

In the diagram, which represents a road-cutting, the ratio 
of ES to AE, or of FS' to BE, is the ratio of slope. 

In practice, the slope at which earthworks are allowed to 
stand vary from 1 to 1, or 45° (as in very coarse material), to 2 
to 1, or 26° 34', in very fine sand. 

A slope of li to 1 (33° 41') is found to be so far suitable 
for all ordinary excavations or embankments, that it is common, 
in the absence of an examination of the material, to adopt it 
as the ratio of slope throughout. 

Setting Slope Stakes. 

317. It is evident that the width of natural surface of 
ground, required in the construction of a road, will vary with 
the depth of excavation or embankment. 

As often, therefore, as it is found necessary to determine 
the depth of the cutting or filling, in the section level, it is 
also necessary to mark the boundaries of the width of the 
work, on the natural surface. This is done by stakes called 
Slope Stakes, and the field-work necessary to determine their 
position, and to measure the section taken across the road of 
which the Slope Stakes indicate the boundaries, is called " Cross- 
Section Leveling," or " Cross-Section Work." 

318. A party of five may be usefully employed in setting 
Slope Stakes ; viz., a leveler, rodman, axeman, and two tape- 
men. 

The rod, for cross-section work, is a ruder instrument than 
that employed in the section level. It should be at least fifteen 
feet long, with the feet and tenths plainly marked. It requires 
no target, the leveler himself reading the rod in the act of 
sighting. 

The field-book is ruled as shown below. 



SEC. v.] 



CROSS-SECTIOi^ LEYELIN'G. 



279 



Dist. 


Left. 


Centre 
cutfings. 


Right. 











The left-hand column contains the distances taken from 
the section-level notes. The third column is for the cut or 
fill, corresponding to the distance in the first column ; these 
numbers also being taken from the notes of the section level. 

A filling, it should be remarked here, is designated as a 
minus cutting in the field-notes. 

The second and fourth columns are for the horizontal and 
vertical measurements of the cross-section. 

319. The examples following will illustrate the method of 
measuring the section and recording the notes : 




^ Let Figure 147 represent a section across a road excavation. 
AB being the bed of the road, and 8C8' the line of the natural 
surface. 

The road-bed is supposed to be 16 feet wide, the centre 
cutting 12.4 feet, and the ratio of slope IJ to 1. 

The level being set up and adjusted in a convenient place, the 
rod is first hold by the centre stake at C, and a reading taken. 



280 ELEMENTS OF SURVEYING. [BOOK VII. 

In the present example, the reading is 5.4. The line AB 
forms a convenient datum line, and the height of the instru- 
ment above this line, is evidently 12.4 + 5.4 = 17.8 ft. 

This is noted down, for the moment, on a reversed page 
of the note-book, or on a spare slip of paper ; neither the height 
of instrument or rod readings being matters of permanent record 
in cross-section work. 

It is evident that if the rod be held at different points along 
the surface, and the readings subtracted from tlie ^* height of 
instrument," the remainders will be the heights of these points 
above the datum line AB. These heights are technically called 
cuttings, although iu the case ot ST and S' V, no actual excava- 
tion is proposed. 

The reading at B is supposed to be 5.5. The cut is therefore 
17.8—5.5 = 12.3. The horizontal distance from the centre is 
8 feet. For each cutting there will be a horizontal measurement, 
and these two must be recorded together. 

The form adopted is that of a fraction in which the numera- 
tor is the cutting and the denominator the distance. 

The record of this measurement would be, therefore, J-|^, 
in the column marked ^Heft." The points A and ^, of the 
cross-section are appropriately termed the angles, and as the 
points B and F, directly over them, become new starting-points 
for horizontal measurements, it is important to distinguish them, 
in some way, in the notes. 

(Right and left in the actual survey are determined by the 
direction in which the survey progresses, and in which the 
centre stakes are numbered.) 

A common method is the one adopted in our notes — ^to sub- 
stitute for the number which represents the half width of the 
road, the letter A. 

The hind-chainman now takes his position at B, and 



i 



SEC. v.] ckoss-sectio:n' levelijs'g. 281 

the remaining distances to the left are measured from this 
point. 

A change in the surface-line at // requires notice. The 
reading of the rod 6.2 indicates a cut of 11.6. This, with the 
distance from E., 10 feet, is duly recorded. 

There being no other material change in the surface line 
beyond H, there remains to be determined on this side only the 
intersection 8, of the surface and slope. 

It is found by trial. When found, it is evident that the 
ratio of the distance to the " cut/' must be the same as the ratio 
of slope. In the present example, the distance must be 1} times 
the cut. 

Suppose a trial reading taken at 25 feet out, is 3.2. The 
height, or cut, is (17.8—3.2) = 14.6. 1^- times this is only 21.9 
feet. The distance tried, 25 feet, is too great. 

Suppose a second trial at 22 feet out, with a rod reading 
of 3.8. The cut is 14. 1^ times this is 21. Still too far out. 

A third trial, at 21 feet out, and a reading of 4, gives a cut 
of 13.8 ft. This multiplied by 1^ gives 20.7 feet. The measured 
distance is slightly too great, but in ordinary practice this ap- 
proximation would be considered near enough. The record for 
the slope-stake *S^ would therefore be |^:f.* 

In proceeding from the centre to the right, we find a pomt 
K between the centre and the angle, that requires attention. 

The distance in such a case is taken from the centre instead 
of the angle. CKh 3 ft. and the rod reading 5.3 gives a cut of 
12.5. The reading at the angle-stake F is 5.8, giving a cut 
of 12 feet. 

If the surface-Hne from i^were level, the distance FS' would 
be 12 X 1|^ = 18 feet ; but as the ground descends, the distance 
is less. 

* The readings and distances in this example have been made to correspond to a rise of 



282 



ELEMENTS OF SURVEYING. 



[book VII. 



A trial at 11 ft. with a reading of the rod of 10.4, indicates a 
cut of 7.4. This multiplied by IJ gives 11.1, which is very nearly 
right ; y^^ is therefore the record for the location of S'. 

The completed notes for this cross-section are as follows : 



Diet. 


Left. 


Centre Cut. 


Right. 




13.8 11.6 12.3 
20.7 10 A 


12.4 


12.5 12 7.4 
3 A 11.1 





320. We will now give a similar example, illustrating the 
method of staking out embankments : 






Fig. 


147. 




Dist. 


Left. 


Centre Cut. 


Right. 




— 15.8 —12.8 
23.7 A 


-11.7 


-8.6 —8.4 
A 12.6 





The filling at the centre is assumed to be 11.7 ft., which 
appears in the column of "centre cut," with its appropriate 

sign. 

The reading of the rod, at the centre, as shown by the dia- 
gram, is 7.2. 






SEC. v.] • CROSS-SECTIOK LEVELIN"G. 283 

The sum of the reading and the centre cut (7.2 — 11.7) 
is —4.5, which is the '^height of instrument" referred to 
the line AB. 

The readings at all other points along the line S8' must 
be subtracted from this "height of instrument," as in the pre- 
ceding example. The several remainders are the corresponding 
** cuttings." 

The reading at angle stake Ey 8.3, subtracted from — 4.5, 
gives —12.8, for the cut. 

For the slope stake 8, we will suppose a trial distance of 20 
feet from E, and a rod reading at the trial point of 10.8 feet. 
The cut is therefore — 15.3 ; this multiplied by IJ gives 
22.95.* The trial distance therefore, 20 feet, is not enough. 

A trial of 24 feet out, we will suppose to give a rod reading 
of 11.3 ft., which corresponds to a cut of — 15.8 ft. The ratio 
applied to this, gives for the proper corresponding distance out, 
23.7 ft., which is nearly correct. The distance at which the trial 
was made is slightly too great. It is evident that if the slope 
stake be set at the calculated distance, 23.7 ft., the record of 

— — ^ may be made without involving an error of more than 

/vO. i 

a tenth of a foot, in either cut or distance. 

On the right, the reading at the angle stake F is 4.1. The 
cut, therefore, is —8.6 feet. 

As the surface rises but little from F to S' the trial distance 
for the slope stake is taken at 12 feet ; (it should be 12.9 ft., if 
the surface were level) ; the reading is supposed to be 3.9 feet. 
This gives the cut —8.4, which should correspond to a distance 
out of 12.6 feet. It is evident, that considering the rise in the 
surface and the rod readings at F and 8', the rod reading at ;S^' 

* The sign is disregarded in the product. It may be well to notice, however, that the 
ratio of slope in embankments is considered to be IJ to —1. 



284 ELEMENTS OF SURVEYING. [BOOK VIL 

would not vary a tenth if moved from 12 feet to 12.6. The 

— 8 4 
record, therefore, for S' is — tttt^- 

i/c.b 

Note. — It is not necessary to take rod readings at the angle 
stakes. If desired, these readings can be omitted and rod read- 
ings taken at the centre stake, and right and left from it at each 
change of inclination of surface. It is often convenient, how- 
ever, particularly for beginners, to take the angle stake readings, 
and hence in the rule these readings are assumed to be taken. 

321. The rules for conducting and recording cross-section 
work, whether for excavation or embankment, are as follows : 

I. Prepare the fleld-hooh by riding columns for Dis- 
tances and Centre Cuttings, leaving wider spaces on each 
side of the latter column for the record of the various 
measurements to the left and right of the centre stake. 
Transfer from the section-level notes the distance and 
corresponding cut or fill, for each stalce of that survey. 
Filling in the cross-section notes is designated as minus 
cutting. 

II. Having set the level in convenient proximity to a 
proposed cross- section, take a reading of the rod at the 
centre-stake. Add this reading to the centime cutting, 
(regarding the sign of the latter), to obtain the ''height 
of instrument." 

III. Lay off half the ividth of the road-bed each side 
of the centre, and mark the distances, temporarily, with 
stakes. Tlxese are the angle stakes. 

IV. Proceed to take rod readings at the angle stakes, 
and beyond them outward (on a line at right angles to 
the direction of the line of the road), at each change of 
inclination of the surface. Subtract each reading from 



I 



SEC. v.] CROSS-SECTION" LEVELIl^G. 285 

the height of instrument ; the remainder is the cutting, 
or vertical distance of the point -measured, from the pro- 
posed road-bed. 

V. Record each cutting, together ivith its horizontal 
distance from the nearest angle stake, in the form of a 
fraction expressing the ratio of the distance to the cut- 
ting. Each fraction being recorded in its proper column 
either " right " or " left " of the centre. Points between the 
centre and angle stahe, are located by measurements from 
the centre. 

VI. To find the position of the slope staJce : Measure 
off a trial distance from the angle stake, and determine 
the cut as before. Multiply the cut by the ratio of height 
to base of the proposed slope. If the trial distance be 
greater than this product, the assumed point is too far 
out, and vice versa. Repeat the trial until the ratio of 
the distance to the cut expresses the ratio of slope. 

322. The cutting at the angle stake is, in cases of a tolerably 
uniform surface, a good guide to the distance to the slope 
stake. Thus, when the angle cutting of an excavation is 16 
feet and the ratio of slope 1^ to 1, the distance out, for a level 
surface, would be 24 feet ; but if the ground in that distance 
rise 2 feet (and which in practice may be determined pretty 
correctly by the eye), then the horizontal distance must be in- 
creased by something more than IJ- times 2 feet. 

When the surface descends, the estimated distance out, for 
a level surface, should in like manner be diminished. In em- 
bankments the conditions are reversed ; the steeper the rise, the 
shorter the distance out. 

323. The following examples will serve to elucidate the sub- 
ject still further: 



286 



ELEMENTS OF SURVEYING. 



[book VII. 



9.7 10.0 110 




Fig. 148. 



Diet. 


Left. 


Centre Cut. 


Right. 




—4. -4.5 
6. A 


— 4.9 


— 6.1 

A 


-6.4 
4 


-7.4 
11.1 





Figure ].48 represents an embankment cross-section, in 
which, by reason of the small depth of filling, the height of 
instrument is a positive quantity. 

The centre cut is —4.9 ; reading of the rod at the centre, 
8.5 ; the sum of these, or "height of instrument," is 3.6. The 
remaining rod readings are given on the line through the 
instrument. 



324. In the example of the following diagram (Fig. 149), 
the cross-section is partly in excavation and partly in embank- 
ment. The ratio of slope is 2 to 1. The centre cut is 2.4. 
The centre reading is 7.9 ; height of instrument, 10.3. 

The reading at H, is 6.3; at E, 6.1; at S, 5.4. The 
point, K, is easily found in practice, it being that point on 
the surface line where the reading of the rod exactly equals 
the height of instrument. The reading at F is 13.2; and 
at S', 15.8. 

From these readings the cuttings may be found, and the 
notes completed as below. 



SEC. v.] 



CROSS-SECTIOK LEVELING. 



287 




Fig. 149. 



Diet. 


Left. 


Centre Cat. 


Right. 




4.9 
9.8 


4.2 
A 


4.0 
3.8 


2.4 




A 


-2.9 
A 


— 5.5 
11 





325. In the following example, there is a regular rise in the 
surface-line of one foot in eight. The ratio of slope in the 
excavation is to be 1 J to 1 ; height of instrument, 14.2. 

In seeking for the position of the slope-stake S', a distance 
out of 13 feet is tried ; the reading of the rod at the trial 
point is 4.8. 




Fig. 150. 

How does this point compare with the true position of S' ? 

Ans. Not far enough out. 
What is the result of a trial at 16 feet out and a reading 
of 4.4 ? Ans. Too far out. 

9.6 



What is the true cut and distance at S' ? 



Ans. 



14.4 



288 ELEMENTS OF SURVEYING. [BOOK VIL 

49 

Find the position of S. Ans. ^4-. 

Note 1. — It sometimes happens in very hilly sections, that 
it is impracticable to sight to all the necessary points of a 
single cross-section from one position of the level. In such 
a case, it is only necessary to work from the centre as far as 
the surface will permit, then establish a turning-point, precisely 
as in section leveling ; change the position of the level so as 
to proceed with the work, and determine the new height of 
instrument, from which the readings are to be subtracted as 
before. 

Note 2. — The degree of accuracy desirable to be attained 
in setting the slope stake, varies with the kind of earth to be 
" staked out," so that no exact rule can be laid down. 

A principle, in quite general use, permits the stake to be 
set when the calculated distance varies from the trial distance 
by less than a foot. 

The limit of error should never be greater than this, but 
in rock and the harder kinds of earthwork, it should be made 
much less. 



SECTION VI. 

COMPUTATION OF EARTHWORK. 

326. Before the work of construction of a railroad or canal 
commences, the calculation of the earthwork must be com- 
pleted. 

The cross-section levels afford the necessary data. These 
surveys have divided the proposed work into blocks of 100 feet, 
or less, in length, and which are appropriately termed prism oids. 



SEC. VI.] COMPUTATION^ OF EARTHWORK. 289 

Different methods are employed for estimating their cubic con- 
tents. The most accurate, though the most laborious, is the 
prismoidal formula (Leg., Mensuration, page 129), 

vol. = g(^ + ^'4-4if). 

B and B' representing the areas of the end sections of the 
prismoid, M the area of a section midway between them, and 
I the entire length of the solid. 

The principal difficulty in applying this formula lies in 
finding the dimensions of the middle section. 

327. We will show the application of the formula by an 
example of road excavation. 

To simplify the problem, we will suppose such a degree 
of regularity in the ground surface that the angle cuttings 
may be omitted. 

The length is supposed to be 100 feet. The other dimen- 
sions are given in the diagram. 

The areas of the end sections are easily found. It is only 
necessary, in each case, to add together the areas of the trape- 
zoids composing the whole end figure, as represented in the 
diagram, and subtract therefrom the sum of the triangles which 
lie outside the section. The dimensions of these triangles are 
always expressed in the cross-section notes, by the records for 
the slope stakes. 

The area of B is thus found to be 104.8 sq. feet, and of i?', 
116 sq. feet. 

Now, if a section of this prismoid be taken midway between 
the two ends, each of its several dimensions must be an arith- 
metical mean of the corresponding measurements of the end 
sections. Thus, the centre cutting is found to be 5 ft; the dis- 



290 



ELEMEI^TS OF SURVEYING. 



[book Vll. 



tance from the angle to slope stake, on the left, 9.9 ft., ( — ^-^) ; 



(7 V I K \ 
-^-^ — ), &G 




10.8 



"^^: 


""■^ 




^ 


— _, 


6.6 \. 




i S. 




XI 


9.9 


8. 




8. 


3.9 



2.6 M 



Fig. 151. 

The area of if is 111.3 sq. ft. 

Vol. of the prismoid = - x 100 (104.8 + 116 + 445.2), 

= 11100 cuhic feet. 

328. In appljdng the prismoidal formula to an example in 
cv^hich one end section has more given dimensions than the other, 
the calculator is frequently in doubt how he shall average these 
dimensions to obtain the middle section. As a rule, each cut- 
ting of the most irregular section should be averaged with the 
cutting nearest opposite to it in the other section. 

We will illustrate this by an example ; representing the sec- 
tions by the field-notes only. 



SEC. VI.] 



COMPUTATIOK OF EARTHWORK. 



29] 



1 

j Dist. 


Left. 


Centre Cut. 


Right. 


2 

2.60 


17.2 
25.8 
11.6 
17.4 


16.8 

A 
11.2 

A 


16 
2 


15.8 
10.4 


13 
A 
10 
A 


10 8.4 
8 12.6 
8 
12 



The half-width of the road, for which A is given in the notes, 
is to be considered as 8 feet. The length of the prismoid is 
expressed by the difference of the given distances, or 60 feet. 
The dimensions of the middle section are found as follows : 
The centre cut is half the sum of the given similar dimen- 

15.8 + 10.4 -,ow . 
sions, ^ = 13.1 feet. 

11.5 
On the right, the average of the angle cuttings gives —^ ; 

for the next measurement, both cut and distance must be aver- 
aged ; it is, 

ix(10 + 8)=9 



J-x(8 + 12) = 10 



or 



10 



The last term in the upper section must be averaged with 
the last in the lower, thus : 

Jx( 8+ 8.4) = 8.2 S^ 

Jx (12 + 12.6) = 12.3 ^^ 12.3* 

On the left, the measurement — of the upper section, must 

be averaged with the centre cutting of the lower, being nearest 
opposite to that point. We have, 

^x (16 + 10.4) = 13.2 13^ 

i.x( 2+ )= 1 ^"^ 1 ' 

14 

At the angle, in like manner, we have -j-; and finally at 

14 4 
the slope stake j^r^- The complete dimensions being 



292 



ELEMENTS OF SURVEYIKO. 



[book VII. 



14.4 14 13^ 
2176 A T~ 



13.1 



11.5 9 8.2 



10 12.3 



The area of the upper section, after subtracting the triangles, 
as before, is 543.47 sq. feet. 

The area of the lower end section is 325.44. 
The middle section contains 429.8 sq. feet. 
The volume of the prismoid is 

I X 60 (543. 47 + 325. 44 + 4 x 429. 8) 
= 25881.1 cubic feet, or 958.56 cubic yards. 



329. The following method* of computing the contents of 
consecutive volumes between regular cross-sections of excavation 
and embankment, contained by uniform slopes, will be found of 
service. 

The accompanying diagram represents the cross-section of a 
railroad cut, h being half the width of road-bed, c centre-height, 



•/""/ I 



V 



h 
Fig. 152. 



r elevation of right slope-stake above grade, r' its horizontal 
distance from nearest side of road-bed, I elevation of left slope- 
stake, V its horizontal distance from nearest side of road-bed, and 



* " Formulae for Railroad Earthwork," by J. Woodbridge Davis, C. E., Ph. D. 



SEC. VI.] COMPUTATIOK OF EARTHWORK. 293 

w the entire top-width or horizontal distance between slope- 
stakes. The area of this section is evidently 

^rh + ilb + ^c {r' + J)) + ^c {V -{-h). 

r I 
Let 8 denote the ratio of slope ; then S = — = j,^ ^nd 

r = Sr', I = SI'. Substituting and reducing, 

Area Section = ^Sb {r' -f V) + ic (r + Z' + 2^). 

Adding and subtracting S/y^ does not change its value ; 

.-. Area Section = ^Sb {r' -\-l' + 2b) — Sb'^ + ic (/ + r + 25), 
or Area Section = ^lo {c-\-Sb)—Sb\ 

Supposing w', d to represent width and centre at next station, 
the area of its cross-section may be expressed by a formula similar 
to the above; half the sum of these, multiplied by the distance, 
/>, between, and divided by 27, gives a near approximate of the 
volume in cubic yards ; 

Vol. = ltvc-\-'w'c'-^Sb {w-{-w')—ASb^] y^. 

Add two consecutive volumes of equal length by means of the 
general formula, w", d , representing the width and centre of 
third cross-section : 

Vol. = [wc + 22dd + w"c" + Sb(w + 2id + iv") - SSb^] ■^. 

By continual addition we may get a formula for the sum of 
any number of consecutive volumes ; but, letting n denote the 
number of volumes, we may at once indite a general formula for 
the calculation of any number of volumes consecutive. Thus 
we have 

^^'" — \ j^Sb(iv-\-2w'-{-&c.-\-2w,^io,+,) - 4:Sb^n ] 108* 

Divide and multiply by 2 to convert the formula into more 
convenient shape, call w'dj w"d', &c., mid-products, wc and 
Wn+iCn+i end-products, and we have 



294 



Vol = 



ELEMENTS OF SURVEYIi^G. 

mid-products + ^ e7id-products 

-]-Sb {mid-2vidtlis-\-^ end-icidths) 

— 2Sb^ X no. of vols. 



[book vil 



Let us illustrate this formula by applying it to the following 
extract from a field-book, containing columns of stations, centre 
cuts, left and right heights and distances of slope-stakes, the 
road-bed being 18 feet, slope 1 to 1, distance apart of stations, 
100 feet : 



station. 


Left. 


Centre. 


Right. 


1 


A^T 


3.0 


■h\ 


2 


A^ 


5.1 


m 


3 


1 oO 


6.4 


^h 


4 


81 
ni 


7.2 


T^.V 


5 


m 


9.0 


^h 


6 


16? 


6.7 


T«A 





OPERATION 






Stations. 


Widths. 




Centre. 




Products. 


1 


11.55 


X 


3.0 


— 


34.65 


2 


. 27.8 


X 


5.1 


= 


141.78 


3 


31.5 


X 


6.4 


=: 


201.60 


4 


34.1 


X 


7.2 


z= 


245.52 


5 


38.0 


X 


9.0 


z= 


342.00 


6 


15.85 


X 


6.7 


= 


106.195 


9 ov Sbx 16S.S0 








1071.745 












1429.2 












-810. 



9)169094.5 
6)18788.3 



3131.38 cu. yds. 



SEC. VI.] COMPUTATION OF EARTHWORK. 295 

330. The foregoing method of calculating earthwork is 
approximate. To find the true contents we use a formula of 
correction, which is obtained in the following manner. Let 

^w {c + Sb) - 8h\ iw' (c + Sb) - SI^ 

be the end areas of a volume of earthwork. Then ^ (tu + w'), 
i{c + c') are evidently the width and centre of mid-section, and 
its area is 



iiw-^w') (^^ + Sb)-Sd^. 



Multiply this by 4, add thereto the end areas, multiply all by ^ 
D, and divide by 27, to obtain in cu. yds. 

True vol - P (^^ + e^V) + (?..' + ^^/.)\_^^ 
irue VOL - y _^^sb{w + w')-12Sb^ 7 324 

This formula would prove very unwieldly to carry through the 
calculations for series of volumes, especially in the consideration 
of intermediate stations. The same results may be obtained in a 
simpler manner by using the difference between true and 
approximate contents as a correction. Subtracting the approx- 
imate volume, 

[wc-\-w'c'-}-Sb(w-}-w')—4:Sb^]^-s, 
from the true, we have, after reduction, 

{w—w'){c'-c)D 
324 ' 

for the error or correction. 



296 ELEMENTS OP SUEVEYING. [BOOK VII. 

^^^- ^^^' We see by the formula that to correct a volume, 

987 

the difference of widths, found by a subtraction in 

— 481 

one direction, must be multiplied by the difference 
208 

of centres resulting from a subtraction in the 

opposite direction, this product multiplied by the 

1449 

length of volume and divided by 324. Applying 

12 ) 3827 this rule to the second volume of example, we 

9 ) 319 have width at station 2 [27. S]—2uidth at station 3 

3)35.4 [31.5] =—3.7; centre station 3 [6A]— centre sta- 

Zu^ Cor. ^^^^ ^ t^-^] = ^•^- -^•'^ X +^-^ 

3131.38 Approx. con'ts. ^^^^ = -^^^' ^^^ ^^'' ^^ ^^ 

«.^« ^~ rr. ,, extra column. Treat each volume 

3119.57 True con ts. 

in like manner, remembering that 

the first and last numbers in column of widths are half-widths. 
We here represent the column of corrections. The sum, — 3827, 
divided by 324, is —11.81 cu. yds. This added to the approx- 
imate result yields for the true answer 3119.57 cu. yds. 

It may happen, in practice, that some of the volumes involved 
need no correction whatever, which fact will be apparent from 
the formula by inspection and without actual labor. 



BOOK VIII. 

TOPOGRAPHICAL SURVEYING, 



331. Besides the surveys that are made to determine the 
area of land and the relative positions of objects, it is frequently 
necessary to make minute and careful examinations for the 
purpose of ascertaining the form and accidents of the ground, 
and to make such a plan as will distinguish the swelling hill 
from the sunken valley, and the course of the rivulet from 
the unbroken plain. 

This branch of surveying is called Topography. In surveys 
made with a view to the location of extensive works, the de- 
termination of the slopes and irregularities of the ground is 
of the first importance ; indeed, the examinations would other- 
wise be useless. 

332. The manner of ascertaining these irregularities is, to 
suppose the surface of the ground to be intersected by a 
S3^stem of horizontal planes at equal distances from each other ; 
the curves determined by these secant planes, being lines of 
the surface, will indicate its form at the places of section, 
and, as the planes are nearer or more distant from each other, 
the form of the surface will be, more or less, accurately ascer- 
tained. 

If such a system of curves be determined, and then pro- 
jected or let fall on a horizontal plane, it is obvious that the 
curves on such plane will be nearer together or farther apart, 
as the ascent of the hill is steep or gentle. 



298 ELEMENTS OF SUKVEYING. [bOOX VIII. 

If, therefore, such intersections are made, and the curves so 
determined are accurately delineated on paper, the map will 
give such a representation of the ground as will show its form, 
its inequalities, and its striking characteristics. 

333. The subject divides itself, naturally, into two parts : 

1st. To make the necessary examinations and measurements 
on the field ; and, 

*^d. To make the plot or the delineations on paper. 

When the area is extended and the contour planes are widely 
separated, points along the ridges and summits, and also along 
the valley bottoms, are located by the Transit or Plane Table, 
and their heights above a datum plane are determined by means 
of Transit angles of inclination, or by the Y Level, or, in rapid 
reconnaissance, by the Barometer. 

We shall, perhaps, be best understood, by giving an example 
or two, and then adding such general remarks as will extend 
the particular cases to others that may occur. 

EXAMPLE FIRST. 

334. Let A, Fig. 153, be the summit of a hill, the contour 
•of «vhich it is required to determine and represent. At A, let a 
stake be driven, and let the axis of the transit, or level, be 
placed directly over the nail which marks its centre. From A, 
measure any line down the hill, as AB, using the telescope of 
the transit, or level, to arrange all its points in the same 
vertical plane. Great care must be taken to keep the measuring 
chain horizontal, for it is the horizontal distances that are re- 
quired. At different points of this line, as a, h, c, d^ &c., let 
stakes be driven, and let the horizontal distances Aa, ai, he, 
and cd, be carefully measured. In placing the stakes, reference 
must be had to the abruptness of the declivity, and the accuracy 



BOOK VIII.] 



TOPOGRAPHICAL SURVEYING. 



299 




with which the surface is to be delineated ; their differences of 
level ought not to exceed once and a half, or twice, the distance 
between the horizontal planes of section. 

Having placed stakes, and measured all the distances along 
the line AB, run another line down the hill, as AC, placing 
stakes at the points e, f, g, and h, and measuring the horizontal 
distances Ae, ef, fg, and gli, Eun also the line AD, placing 
stakes at i, I, m, and n, and measuring the horizontal distances 
Ai, il, Im, and mn. 

Each line, AB, AC, AD, running down the hill, from A, 
may be regarded as the intersection of the hill, by a vertical 
plane ; and these secant planes are to be continued over all the 
ground which is to be surveyed. If the work is done with a 
transit, or with a level having a compass, the angles DAB 
and BAC, contained by the vertical secant planes, can be 
measured ; if it is done with a level, having no needle, let any 



300 



ELEMENTS OF SURVEYIN'G. 



[book VIIL 



of the distances ae, If, ai, hi, &c., be measured with the chain, 
and there will then be known the three sides of the triangles 
Aae, Ahf, Aai, AN, &c. 

Let, now, the difference of level of all the points marked in 
each of the lines AB, AD, AC, be determined, being careful to 
hold the rod upon a point near each stake which represents 
the general surface, and not at the bottom of a hole nor 
upon the top of a mound, a precaution to be observed in every 
leveling operation. 

Let now the heights of all the points marked on each of the 
lines AB, AD, and AC, be found with reference to the datum 
plane, which, near the coast, may be the plane of mean low 
water, or a plane assumed so that it will lie below the lowest 
point to be delineated. 

In the present example, of a single slope only, the results of 
the measurements and leveling are : 



Line AB. 



Distances. 
Aa = 4:0 feet. 
ab = 50 " 
be = 30 " 
cd = 4:6 '' 



Heights above datum plane. 
A = 64: feet. 
« = 52 " 
^ = 44 " 
c = 35 " 
d = U " 



Distances. 
Ae = 28 feet. 
ef = 45 '' 
fg = 55 " 
gh = 49 



<( 



Line AC. 



Heights above datum plane. 
A = 64: feet. 
e = 53 " 
/ = 44 " 
^r = 32 " 
h = 18 



(t 



BOOK VIII.] TOPOGRAPHICAL SURVEYING. 301 

Line AD. 
Distances. Heights above datum plane. 

Ai = 25 feet. ^ = 64 feet. 

il = 55 " i = 55 '' 

Im = 38 " I = 4:2 '' 

mn = 48 " m = 35 '' 

n =z 21 '' 
Angle CAB = 25°, Angle DAB = 30°. 

These data are sufficient, not only to find the intersections 
of horizontal planes with the surface of the hill, but also for 
delineating such curves of section on paper. 

Plot of Work. 

335. Having drawn, on the paper, the line AB, lay off the 
angle BAG =25°, and the angle BAD = 30°. Then, from a 
convenient scale of equal parts, lay off the distances Aa, ab, be, 
cd, Ae, ef,fg, gh, Ai, il, Im, and mn. 

Let the horizontal planes be passed at distances of 8 feet 
apart, in which case the point A, in the example given, will 
lie in the eighth contour plane, counting from the datum 
plane. Since A is the highest point of the hill, and the 
difference of level of the points A and «, is 12 feet, the first 
plane, reckoned downwards, will intersect the line traced on 
the ground from A to B, between A and a. Regarding the 
descent as uniform, which we may do for small distances, without 
sensible error, we have this proportion : as the difference of level 
of the points A and a, is to the horizontal distance Aa, so is 8 
feet, to the horizontal distance from A to where the first hori- 
zontal plane will cut the line from A to B. This distance being 
thus found, and laid off from A to o, gives o, a point of the 
curve in which the seventh plane intersects the ground. The 
points at which it cuts the line from A to C, and the line from 



302 



ELEMENTS OF SURYEYIXG. 



[book VIII. 



A to D, are determined similarly, and three points in the seventh 
curre are thus found. 

The graphic operations are greatly facilitated by the aid of 
a sectoral scale of equal parts, of which Fig. 154 is a repre- 
sentation. 




Fig. 154. 



It consists of two arms, or sides, which open by turning 
round a joint at their common extremity. 

On each arm of the sector, there is a diagonal line that 
passes through the point about which the arms turn : these 
diagonal lines are diyided into equal parts. 

The advantage of the sectoral scale of equal parts, is 
this : 

Let it be proposed to draw a line upon paper, on such a 
scale that any number of parts of the line, 40 for example, 
shall be represented by one inch on the paper, or by any part 
of an inch. Take the inch, or part of the inch, from the scale 
of inches, on the sector ; then, placing one foot of the dividers 
at 40, on one arm of the sector, open the sector until the 
other foot reaches to the corresponding number on the other 
arm ; then, lay the sector on the table without varpng the 
angle. 

Now, if we regard the Hues on the sector as the two sides of 
a triangle, of which the line 40, measured across, is the base, it 



BOOK YIII.] TOPOGRAPHICAL SURVEYIl^G. 303 

is plain, that if any other line be likewise measured across the 
angle of the sector, the bases of the triangles, so formed, will be 
proportional to their sides. Therefore, if we extend the dividers 
from 50 to 50, this distance will represent a line of 50, to the 
given scale ; and similarly for other lines. 

In the example before ns (^ig. 153), let it be borne in mind, 
that the descent from A to a, is 12 feet, and that it is required, 
upon the supposition of the descent being uniform, to find that 
part of the distance corresponding to a descent of 8 feet. Take 
the distance from A to a, in the dividers, and open the arras of 
the sector until the dividers will reach from 12 on the line of equal 
parts, on one side, to 12 on the line of equal parts on the other. 
Then, without changing the angle, extend the dividers from 8 on 
one side to 8 on the other ; this will give the proportional 
distance to be laid off from A to o. Or, if the dividers be 
extended from 4 to 4, the proportional distance may be laid off 
from a to o. 

If the distances to be taken from the sector fall too near 
the joint, let multiples of them be used. 

336. The sixth plane is to pass 8 feet below the first, 
that is, 16 feet below A, or 4 feet below a, a being 12 feet 
below A. Take the distance ai, in the dividers, and extend 
the sector, so that the dividers will reach from 8 to (the 
descent from a to b being 8 feet) 8, or from 80 to 80 ; then, 
the distance from 4 to 4, or from 40 to 40, being laid off 
from a to p, gives p, a point of the sixth curve. 

The difference of level between a and b being 8 feet, and 
the difference of level between a and p being 4 feet, the dif- 
ference of level between p and b must also be 4 feet ; hence, 
the fifth plane will pass 4 feet below b, and q, determined 
as above, is a point of the fifth curve, and so on. After 
having determined the points in which each contour line cuts 



304 



ELEMEIiTTS OF SUBVEYING. 



[book VIII. 



the lines diverging from A, let the contour lines be drawn 
through them, so as to indicate the surface of the hill. The 
numbers (64), (56), &c., show the vertical distances of the 
respective planes above the plane of reference. 

337. Having drawn the horizontal curves, the next thing to 
be done is so to shade the drawing that it may represent accu- 
rately the surface of the ground. This is done by drawing 
a system of small broken lines, as in the figure, perpendicular 
in direction to the horizontal curves already described. In 
all topographical representations of undulating ground, the 
lines of shading are drawn perpendicular to the horizontal 
curves, to indicate the direction of the flow of water down the 
declivity. 

A profile along either of the diverging lines may be plotted 
by the rules already given. Fig. 155 shows the profile along 
the line AB, 




B d 



EXAMPLE SECOND. 

338. The following example will illustrate the methods em- 
ployed in making a topographical survey, where great accuracy 
is required. 

By means of a transit or level, range a line of stakes A, B, C, 
D, E, &c., Fig. 156, along one side or through the middle of the 
ground to be surveyed, at equal and convenient distances from 
each other, say 100 feet apart. Mark, with a piece of red chalk, 
on each stake in this row, one of the letters of the alphabet, 



A 



BOOK VIII.] 



TOPOGRAPHICAL SURVEYING. 



305 



A, B, G, D, E, &c., in their order. At A, range a line of stakes, 
perpendicular to AE, planting the stakes at intervals of 100 
feet; and mark them with the letters Aq, A^, A^, &c., which are 
read A zero, A one, A two, &c. 



D 



E 



A. 




1 


A. 

2 


A A 
3 4 





B 

1 


B 

z 


B B 

3 4: 


C 




c 

1 


C 

2 


C C 

3 4 


D 



Eo 


D 
\ 

El 


D 
2 

a 

E2 


D D 

3 4 

E3 Ea 



Fig. 156. 



At ^, range a line of stakes also perpendicular to AE, and 
at distances of 100 feet from each other, and designate them 
B^, B^, ^3, &c. Do the same at (7, D, E, &c., until all the stakes 
are placed, dividing the area to be surveyed, into squares of 100 
feet on a side. The letters and figures should be plainly marked 
on a smooth face of each stake, for facility of reference. If this 
system of notation be followed, the stakes may be recorded 
without danger of confusion. 

339. The following is the form of a field-book, used in topo- 
graphical leveling : 



306 



ELEMENTS OF SURVEYIKG. 



[book VIII. 



Field Notes. 



Benches. 


+ Sights. 


Height Inst. 


— Sights. 


stations. 


Heights. 


Bench 


4.136 


12.142 


12.1 


E. 


0.0 








1.9 


D. 


10.2 


C3 


11.906 


22.441 


1.607 


Ca 


10.535 








6.0 


E. 


16.4 








6.9 


Es 


15.5 








6.8 


D3 


15.6 








1.9 


Di 


20.5 








3.7 


0. 


18.7 








6.8 


B4 


15.6 








1.7 


C. 


20.7 


B3 


11.914 


33.448 


0.907 


Bs 


21.534 








8.0 


E„ 


25.4 








4.0 


B. 


29.4 








4.1 


D„ 


29.3 








1.9 


c. 


31.5 








5.0 


A3 


28.4 








9.9 


A. 


23.5 








1.7 


I>4 


31.7 








1.1 


E4 


32.3 








0.1 


A. 


33.3 


Co 


11.813 


45.225 


0.036 


Co 


33.412 








4.8 


B. 


40.4 








2.6 


Bo 


42.6 


Ai 


8.925 


52.669 


1.481 


A, 


43.744 








3.2 


Ao 


49.5 



340. In the example taken for illustration the point E^ is the 
lowest point. 

Set up the level and take a reading upon a bench, which has 



BOOK VIII.] TOPOGRAPHICAL SURVETIKG. 307 

been determined to be 8.006 feet above some plane of reference, 
the sea level for instance. Suppose the reading to be 4.136, 
which is entered in the " + Sight " column. The " Height of 
Instrument" will therefore be 12.142, as entered in the proper 
column. Next take readings upon E^y Dc^, and Cg, reading C^ 
to thousandths as it will be used as a "turning point" (Art. 304). 
Kemove the level to a new position and read C^ again, 11.906, 
which gives a new " Height of Instrument " 22.441. From this 
position read as many stations as possible, and then use B^ as a 
" turning point," and so on. 

Plotting the "Work. 

341. Draw, on a piece of paper, a straight line AE. From 
a scale of equal parts, set off distances AB, BC, &c., each to 
represent 100 feet. Erect perpendiculars to AE, at each of the 
points A, B, G, &c., and then set off the distances from A to 1, 
from 1 to 2, &c., each to represent 100 feet ; and through the 
points 1, 2, 3, and 4, draw parallels to AE. These, by their 
intersections with the lines drawn through A, B, C, &c., will 
determine the position of the stakes Aq, Ac,, &c.; and write in 
red ink on the plot, the height above the plane of reference of 
each stake, taken from the column of total differences in the 
field-book. Let us suppose that the horizontal planes are to be 
taken at distances of 6 feet. We may find the points in which 
the contour lines intersect the sides of the rectangles, as in 
Example First. A very convenient scale for finding the points 
in which the contour lines cut the sides of the rectangles 
may be constructed thus: Upon any line as AB, Fig. 157, 
erect equidistant perpendiculars as at 0, 1, 2, &c. Parallel 
to AB draw lines, alternately heavy and light, as at 1, 2, 3, &c. 
Suppose we wish to find where the 12-foot plane cuts the side of 
a rectangle, C^G^ for example. The height of G^ is 10,5, and 



308 



ELEMEN^TS OF SURVEYING. 



[book VIII. 



of G^ it is 20.7, from which data we find the rise from C^ to G^ 
to be 10.2. The rise from G^ to the 12-foot plane is 1.5. 

























13 
12 
U 






























































































































10 " 


-r 


















































/' 


9 
8 

•J 












































/ 






















,/-'' 








6 
.S 
4 
3 
2 
1 




























































/' 












































S/-'- 


— 


















A 


/ 


j 




















12 3 


4 


15 6 9 


f 8 9 1 



















F 


[Q. 1 


57. 













Now fasten a fine thread at A on the scale, and stretch it to 
cut 10.2 on the vertical scale, as ^C in Fig. 157. Look along 
the horizontal line 1.5, as dotted, and its intersection with AG 
will be at 8, distant 16 ft. on the horizontal scale from A, or G^. 
The tenths on the vertical and horizontal scales may be 
estimated by the eye with sufficient accuracy. Engineer's 
Section Paper may be advantageously used for the above 
purpose. 

342. If only a rough plot is needed, the Surveyor may take 
the plot thus commenced, into the field, and by the eye trace 
the contour lines on the map. If we note where the lines of the 
rectangles cut fences, roads, streams, &c., we can, by joining the 
points, obtain a plot of the ground. 



BOOK VIII.] TOPOGRAPHICAL SURVEYmG. 309 

The coutour lines may be traced on the ground as follows : 
Set up the instrument and read a staff placed upon a bench, and 
determine the height of instrument above the datum plane. Sup- 




2a5 



15.6 



20J' 



31.7 



32.9 



Fig. 158. 



pose it to be 11.432. If we wish to mark out the contour line six 
feet above the datum plane, we set the target of the leveling rod 
at 11.432—6 = 5.432, and direct the rodman, by signals, up or 
down the hill, till the horizontal wire of the telescope coincides 
with the horizontal line of the vane. The foot of the staff is 
then 6 feet above the datum plane. Let a stake, marked 6, be 
driven here, and direct the rodman around the hill, until a 
second position shall be found, when the horizontal wire of the 
telescope will cut the vane, and drive there another stake, 
marked 6 ; and so on, until a sufficient number of stakes have 
been driven to determine the curve (6). Then, let the line of 
stakes, marked 6, be surveyed with the compass and chain, and 
plotted. Other contour lines may be found in a similar manner. 



310 



ELEMENTS OF SURVEYING. 



[book VIII. 



343. A practical application of this method is also called for 
when a surveyor is required to determine, in advance, the area 
of land which will be submerged by the construction of a dam. 
lu this case the contour plane is fixed by the proposed height of 
the dam. Having set up at any point, a reading is taken upon 
the rod held upon a point at the proposed height of the dam, and 
the rodman is then directed to points along the shore of the 
proposed pond, or reservoir, which shall give the same reading. 
In this way the contour of the pond, when there is no overflow at 
the dam, can be determined. If the calculated depth of water 
upon the dam at the maximum overflow be added to the rod 
reading used above, the high-water contour may be also traced 
out. 

344. When the plane of reference is so chosen that points 
of the work fall on different sides of it, all the references on one 
side are called positive, and those on the other, negative. The 
curves having a negative reference are distinguished by placing 
the minus sign before the number ; thus — ( ). 



Shading and Delineation. 

345. Fig. 159 represents a piece of ground sloping towards 
i>, which is the lowest point ; and through this point the plane 
of reference is supposed to pass. The following table indicates 
the heights of the several points above the plane of reference. 





Ft. 


Ft. 




Ft. 


Ft. 


c above 


D, 2 


H above D, 7 


V 


above D, 9 


B about D, 12 


d " 


D, 4 


h '' D, 1 


q 


" D, 9 


L '' D, 13 


}i '' 


D, 4 


s '' D, 1 


c 


'' D, 9 


'' D, 14 


t 


A 4 


f '^ D,% 


n 


'' D, 11 


a ^^ D, 15 


9 '' 


D, 5 


I '' A 8 


i 


" D, 12 


F '' D, 15 


I '' 


D, 5 


I " Z), 9 


m 


'' D, 12 


E '' D, 17 



A above D, 20 feet. 



BOOK vrii.] 



TOPOGRAPHICAL SURVEYING. 



311 



The first horizontal plane is passed 2 feet above D, and the 
curve of intersection with the surface passes through c. The 
second secant plane is passed at 3 feet above D, and intersects the 



i./liiiiiiii'i:iiN!ntm 




surface, in the curve uvy and also near d, which is one foot above 
the curve. All the other secant planes are passed at three feet 
from each other ; and, comparing the height of each point above 
D, with the curves lying nearest, on either side, the positions of 
all the points, with respect to the curves, and with respect to 
each other, are easily seen. 

346. The manner of shading the map, so as to indicate the 
hills and slopes, consists in drawing the lines of shading per- 
pendicular to the horizontal curves, as already explained (Art. 
337). These shading lines are drawn close together, when the 
slope is abrupt, and further apart, as it grows more gentle. 
Figure 159 indicates the method of shading. 

347. In topographical surveys, great care should be taken to 
leave some permanent marks, with their levels written on them 



312 ELEME^^TS OF SURVEYIXG. [BOOK VIII. 

in a durable manner. For example, if there are any rocks, let 
one or more of them be smoothed, and the vertical distance from 
the plane of reference marked thereon ; or let the vertical 
distance of a point on some prominent building be ascertained 
and marked permanently on the building. Such points should 
also be noted on the map, so that a person, although un- 
acquainted with the ground, could by means of the map go 
upon it and trace out all the points, together with their differ- 
ences of level. 

348. Besides representing the contour of the ground, it is 
often necessary to make a map which shall indicate the woodland, 
the marsh, roads, ditches, etc. For this, certain characters, or 
conventional signs, have been agreed upon, as the representatives 
of things, and when these are once tixed in the mind, they 
readily suggest the objects for which they stand. Those which 
are given in the four following pages have been adopted by the 
TJ. S. Coast and Geodetic Survey, and are used in all plans and 
maps made under its direction. 

It is very desirable that a uniform method of delineation 
should be adopted, and it is, therefore, recommended that the 
conventional signs given in the accompanpng plates be carefully 
studied and uniformly followed. 



BOOK VIII.] 



TOPOGRAPHICAL SURVEYING. 



313 




PiQ. 160.— Sparsely settled Town, Salt Marsh, Pine Woods, Ditches, Fences, and Undefined 

Roads. 



^^^^^Ml 









lii^JfT_- 






SM"^ 



g£L 



Fig. IQl .— Blockiug of Cities, Large Buildings, Suburban Vilius and Grounds, Fresh Marsh. 



314 



ELEMEN"TS OF SURVEYING. 



[book VIII. 




Fig. 162.— Eailroads, Canals, Iron bridges, Eocky-cliffs, Mid-river drift, Water-worn Rocks. 

Mixed Woods over hill curves. 




Fig. 163.— Eroded drift banks, with boulders set free ; and scrub deciduous woods. 



BOOK VIII.] 



TOPOGRAPHICAL SFRVEYIN"G. 



315 




Fig. 164.— Heavy Oak Woods, Reclaimed Marsh, and Orchards. 




Fio. 165.— Fresh Water Pond, Meadow Grass, Sage Brush, and Arroyos. 



316 



ELEMENTS OF SURVEY! XO. 



[book VIII. 







Fig. 166.— Sand and Shingle Beaches, Eroded Earth Banks, Roads, Fences, Shaded Road- 
sides, Hill-shading. 




BOOK IX. 

RAILWAY CURVES. 



M 



349. After the route of a railroad or canal has been deter- 
mined by reconyiaissance, the centre line of the work is estab- 
lished by a transit (see Section III, B. IV). 

350. This preliminary survey establishes a succession of 
straight lines, of greater or less length, according to the obsta- 
cles to be avoided or the advantages to be gained, arising from 
the nature and the contour of the ground. 

The angle formed, at each change in the direction of the 
route, is carefully measured and recorded. 

In the final survey or location, these angles are replaced 
by curves ; and in order that the change in direction shall 
be as gradual as practicable, the straight lines of direction 
are made tangents to the curves at their point of meeting. 

The preliminary survey is termed, by the engineer, " running 
out tangents." 

351. We will proceed to describe the method of locating 
curves, first giving the mathematical principles applicable to 
the subject. 

Let AD and DB, Fig. 168, be two tangents to the arc of a 
circle, AB. Draw the radii AC, BC, and the secant CD. 

The following relations are easily deduced. The tangents 
AD and DB are equal (Leg., Bk. Ill, Prob. 14). The angles 
A and B are right angles (Leg., Bk. Ill, Prop. 9), consequently 
the angles C and D, of the quadrilateral ADBC, must be sup- 



318 



ELEMENTS OF SUKVEYING. 



[book IX. 




(1) 



plements of each other. The angle TDB, therefore, must be 

equal to the angle ACB. The 

right-angled triangles ADC and 

BDC are equal (Leg., Bk. I, 

Prop. 17) ; hence, the angle 

DC A is equal to DCB, and 

each equal to ^TDB. 

Let the radius AC hQ rep- 
resented by r, the distance AD 
by d, and the angle TDB 
by a. Then will (Trig., Art. 
66), 

d =z r tang ^a. 

The angle TDB is the angle formed by two straight lines 
of the preliminary survey, and is carefully measured by the 
engineer, in locating tangents. 

From formula (1), we can determine the value of d, for 
any given values of a and ?•; and hence we can determine 
at what point on the tangent, laid off from i>, the curve of 
any given radius must commence. 

It is evident, both from the diagram and the formula, that 
for any given angle between the tangents, the greater the 
radius of the curve, the greater will be the distance cut off 
between the intersection of the two tangents and the point of 
tangency. 

It is sometimes necessary to give a particular value to d* 
In such case, we use the formula, 

r = d coi \a. (2) 

352. The work of laying out or locating a curve in the 
field is somewhat simplified, if the curve have such dimensions 
that one chain, of 100 feet, have an arc corresponding to an 
exact number of degrees. 



BOOK IX.] KAILWAT CURVES. . 319 

The radii of such curves are easily calculated. Thus, a 
circle in which one degree of arc measures one chain, will 
have a circumference of 360 chains, or of 36,000 feet, and con- 
sequently, a radius of -r — —-t::-^. = 5729.58 feet. 

In a circle in which two degrees of arc correspond to a 
chain, the radius will be only half as great, or 2864.79. 

When three degrees of arc measure one chain, the radius is 

5!?M8 = 1909.85 feet. 
o 

The number of degrees, corresponding to one chain, of a 
TSiWwaj cur\ef is csilled the 'Ulegree of curvature.'* 

The radius of a one-degree curve is 5729.58 feet; of a two- 
degree curve, 2864.79 feet, &c. 

Representing the degree of curyature by c, we have the 

formula, 

5729.58 ,^, 

r = -^-l (3) 

r being expressed in feet, and c being the number of degrees 
corresponding to 100 feet of arc, or in common practice to 
100 feet chord. 

353. Apply the preceding formulas (1), (2), (3), to the fol- 
lowing examples: 

1. If the angle TDB, of the tangents, be 45° 10', what dis- 
tance must be laid off from the intersection D, to the point of 
tangency, to admit of a 4° curve ? From formula (3), we have, 

r = ^^^^^ = 1432.39 feet. 
4 

Substituting this value of r in formula (1), we have, 
d = 1432.39 tang 22° 35' = 595.76 feet. 

2. If the angle a be 30°, and the distance d be 600 feet, 
what is the radius? Ans. 2239.2 feet. 



320 



ELEMENTS OF SURVEYING. 



[book IX. 



3. What is the degree of curvature in the last example ? 
Formula (3) gives 

5729.58 



C =: 



= 2.°5587 = 2° 33' 31". 



4. The angle a being 20° 21', what is the value of d for 



a one-degree curve ? 



Ans. 



Location of Curves by one Transit. 

354. The location of curves, according to the most common 
method, consists in laying off, at the point of tangency A, such 
angles as shall just subtend one chain of arc. 

If the arcs Av, vw, tux, &c.. Fig. 169, represent arcs of one 
chain each, the angles AGv, vCw, &c., are each equal to the 
degree of curvature. 




The angles DAv, vAw, wAx, are each equal to one-half the 
degree of curvature (Leg., Bk. Ill, Prop. 7, Sch., and Prop. 21). 

The operations in the field are very simple. The party 
should consist of a transitman, two chainmen, and an axe-man. 



BOOK IX.] RAILWAY CURVES. 321 

The transit is set and adjusted at a tangent point, as A, and 
directed along the tangent toward D. 

An angle equal to half the degree of curvature is deflected 
from AD toward the side on which the curve is to run. The 
hind-chainman holds his end of the chain at A. The fore- 
chainman, keeping the chain carefully extended, is directed by 
the transitman into line with the axis of the telescope. This 
locates the point v on the curve. 

From the line Av, another deflection is now made, of the 
same angle as before. The chainmen move forward ; the hind- 
chainman stopping at v, while the fore-chain man, keeping the 
chain extended, is directed by the transitman as before, and a 
second stake, w, is fixed on the curve. 

By continuing the process of deflecting angles equal to half 
the degree of curvature, and causing these angles to subtend 
measured distances of one chain each, the entire curve is 
located. 

355. The last deflection on the curve rarely corresponds to 
an entire chain ; it is, therefore, less than the others. Its amount 
can be readily calculated, when it is remembered that the sum 
of all the deflections, or the angle DAB, is exactly equal to 
one-half the angle a. 

Hence, if from ^ the sum of the deflection angles laid off 

be subtracted, the remainder will be the final deflection angle, 
called the sub-deflection angle. The corresponding sub-chord is 
such part of 100 ft. as the sub-deflection angle is of the deflection 
angle. In practice the sub-chord should always be laid off, using 
the sub-deflection angle, to check the work ; the final peg, thus 
located, should fall upon the previously determined tangent-peg. 

By this method of laying out curves, any error in chaining 
any one chord is carried into all succeeding portions of the curve. 



322 ELEMENTS OF SUEVEYIKG. [bOOK IX. 

356. It is sometimes necessary to remove the transit from 
the transit- point to some other point on the curve, before the 
location has been completed. 

In such a case, the direction of the tangent to this new point 
should be determined. Suppose x, Fig. 169, to be a located point 
Dn the curve to which the transit has been transferred, and 
from which new points beyond x are to be located. Adjust the 
transit and direct the telescope to A. Lay off the angle Axt, 
equal to BAx (the sum of the deflections made in locating 
V, w, and x), — xt is the tangent. By revolving the telescope, 
the tangent is produced to s, from which deflections may be 
made as at first. 

Note 1. — The selection of the radius is governed by cir- 
cumstances. Curves of the longest possible radius are, in rail- 
roads, always the most desirable ; but the larger the radius for 
any particular pair of tangents, the greater the distance by 
which the curve will depart from the intersection of the 
tangents. It may happen, therefore, that too large a radius 
may lead to an obstacle, which the angle in the first survey 
was made to avoid. 

The map, therefore, of the preliminary survey, should include 
so much of the topography of the adjacent section, that the 
radii of the curves may be selected by an inspection of the 
map. 

Note 2. — It will be observed that it is the chord, and not 
the arc, that is measured for each deflection, when locating 
in the field ; the difference, in railway curves, of proper dimen- 
sions, does not lead to sensible error. 

For curves of a short radius (less than two thousand feet), 
the error may be diminished, by locating the stakes at half- 
chain distances, deflecting, of course, half the calculated deflec- 
tion angle. 



BOOK IX.] RAILWAY CUEVES. 3^3 

Location of Curves by two Transits. 

357. The surface, over which it is necessary to locate a curve, 
may be of such a character as to render it impracticable for the 
chainmen to make their measurements ; if, however, the various 
points are accessible to the axeman, as in the case of marshes, 
shallow lakes, or bays, the stakes may be accurately located by 
the simultaneous deflections of two transits. 

The method is based on the following geometrical principle : 




Fig. 170. 

Let A and B be the two tangent points of the curve AvB, 
and D the intersection of the tangents. 

If from any point v, on the curve, the lines vA, vB, be drawn, 
then the sum of the angles vAB and vBA is measured by one- 
half the arc AB, and is therefore equal to one-half the angle a, 
or to either of the entire angles A or B. 

To locate the curve in the field, a transit is set at each of 
the tangent points A and ^, and the deflection angle is deter- 
mined as in the first method. 

The transitman at A deflects, in the iffeual way, one deflec- 
tion angle from the tangent AD. At the same time, the transit- 
man at B deflects the same angle from the chord BA, or what 
amounts to the same, he deflects the difference between this 
angle and -J^a, from the tangent BD. The lines of sight of the 



324 



ELEMEN"TS OF SURVEYING. 



[book IX. 



two telescopes now intersect at a point v, on the curve, one chain 
from A. The flagman, directed at the same time by both 
transitmen, is readily brought to the location of the point. 
By a repetition of this process the entire curve is located. 

Location of Curves by the Chain alone. 

358. It is sometimes convenient to locate a curve without 
using the transit. In such case, the following method is gen- 
erally employed. 

Let A represent the point of tangency, G the centre, and 
V, 10, X, located points of the curve, one chain apart. 

From V, draw vu perpendicular to the tangent, and it will 
be the first offset, which denote by o. Denote the length of the 
chain by ?, and the radius A C, by r. If, now, we suppose A G 
to be prolonged till it meets the circumference in some point, 
on the other side of the centre (7, and this point then to be 




jomecr with v, and vn then drawn parallel to the tangent, we 
shall have (Leg., Bk. IV, P. 23), 



1% 
Av^ = 2r • An ; hence, o = ^ 



(4) 



If, DOW, we prolong Av till vt := Av, and join t and w, 



rv 

BOOK IX.] RAILWAY CURVES. 325 

tw will be the second offset, and will be double vu. For, the 
triangles in the figure, whose vertices are C, and whose bases 
are the equal chords Av, viv, &c., are isosceles and equal. 
Now, in any one of the triangles, the sum of the two angles 
at the base and the vertical angle C, is equal to two right- 
angles. But, since Avt is a straight line, tvw-\-wvC-j-CvA, 
is also equal to two right angles. Therefore, tvtu is equal to 
any one of the equal angles at C, and is, consequently, double 
the angle uAv, which is half the angle C. 

Since the triangle tuvt is isosceles, if vp be drawn perpen- 
dicular to the base, it will bisect both the baseband the vertical 
augle, making tp =piv. But the right-angled triangles Auv and 
vtp are equal (Leg., Bk. I, P. 6); hence, ho = 2vu. Denoting 
the second offset by o', we have, 

c- = ^. (5) 

359. The practice is as follows : Having calculated the first 
offset 0, fix one end of the chain at A, Fig. 171, and stretch the 
chain to v, measuring uv = o perpendicular to the tangent ; the 
measurement of o may be made with a short supplemeiitary steel 
tape, its end division being marked to hundredths of a foot, or 
with a rod marked with two divisions each equal to o. Next 
hold one end of the chain at v and align a pin at t, on Av pro- 
longed, making vt = Av ; then swing the chain towards w till 
the measured distance tw =: 2o :=■ o'. Prolong vw as before and 
find a new point x, and so continue till all the full chords have 
been laid down. The last chord is usually a suh-chord and its 
offset must be calculated separately. Suppose it is required to 
run a curve of 1000 feet radius, the angle of intersection, «, Fig. 

170, being 48°. From the formula d = r tan — we find DA = 

4-45.2, and then locate the tangent peg at A, and also that at B. 



326 ELEMEISTS OF SURVEYING. [BOOK IX. 

We also find o = 5 and o'= 10. The angle at the centre sub- 
tended by the 100-ft. chain is found by the relation L^^ — 

r 

sine \ angle at centre (Leg., Trig., Art. 64). In the case supposed 
we find the angle subtended to be 5° 44'. The number of full 

48° 
chords is equal to o .., -, which gives 8 full chords, and a remain- 
ing angle at the centre of 2° 8'. To find the sub-chord we have 

sine (?-^) X 1000 x 2 = 37.24 feet. 

The final offset is ^^^^^ = 1.39. 

r 

The peg at the end of the sub-chord should fall at the second 

tangent peg, if no errors have been made in chaining. 

Note. — In employing this method of locating curves, the 
aligning by which the chords are produced should be done with 
much care, as any error in locating a stake involves much 
greater and increasing errors in succeeding stakes. 

This is called, by engineers, ** the method by offsetting from 
tangent ^nd chords produced." 

EXAMPLES. 

1. What are the tangent and chord offsets, for a curve of 
2000 feet radius ; the stakes to be 100 feet apart ? 

Ans. From tangent, 2.5 ft.; from chord produced, 5 ft. 

2. Find offsets for a one-degree curve. 

Ans, Tangent, .87 ft.; chord, 1.74 ft. 

360. Let it be required to run out a curve of 500 ft. 
radius, the stations being 25 feet apart on the curve. 

As the chord of the angle i is to be 25 ft, we have (Fig. 172), 



BOOK IX.] 



BAILWAY CURVES. 



327 



i 124 ^„^ 
^^"' 2 = 500 = '^^^ 



from which we find { = 2° 52'. Now, 

Tx :='bs ^^ sine i x 500 = 25 feet ; 
and xb—Ts — 500 — (cos i x 500) = .63 ft. 

Measure from T 25 feet to 
x^ and at x lay off the per- 
pendicular xh = .63 ft., thus 
locating the point b on the 
curve. To locate c we have 

Ty = tvc = sin 2 ^ x 500 
= 49.95 ft., and 

yc=Tw = 500— (cos 2i x 500) 
= 2.50 ft. 



j; 







■ — 


— •- ' 




W^ 


'"^ 






r^ 






/ 






y/ 




/ jfc 








Ji\ 


\ 






l\ 1 


— i i 







s w 



Fig. 172. 



The next tangent distance to z equals sine 3^x500, and the 
offset at z equals 500— (cos 3ix500), and so on for succeeding 
points. 

When the offsets become too long to be readily and correctly 
set out, a new tangent is located thus: Prolong cd to li making 
dh = cd, and then bisect Tie {e having been already located) in/ 
and range a line df, which line will be tangent to the curve at d. 
The correctness of the new tangent should always be checked by 
locating from it the third or fourth station counting back 
towards T. The same computed values that were used in 
locating the points already fixed may be again used in locating 
new points from the new tangent. 



Laying off the Ordinates. 

361. The methods described thus far for locating railway 
curves, apply to points 100 feet apart. This is sufficiently 



328 



ELEMENTS OF SURTETIXG. 



[BOOK IX. 



accurate for the earthwork. In laying the track, however, stakes 
every ten or twelve feet are necessary. These are set by drawing" 
the chain or tape in a straight line between the 100-ft stakes, 
and measuring from it, offsets, as often as desirable, to the 
intermediate points of the curve. 

The length of these offsets, or ordinates, is calculated in 
the following manner : 




Let VW represent a 100-ft. chord of a railway curve, of which 
C is the centre. Draw the diameter HK parallel to VW, and 
drop the perpendicular VL. Then, 

FX2 = EL X LK. 

(Legendre, Bk. IV., Prop. 23, cor. 2). Since 

EL = r — 50, 

and LE = r + 50, 

the value of VL is readily calculated for known yalues of the 

radius. 

Let NM be an ordinate, at any distance from VL, say 

10 feet. Then, 

Xm = EM X MX; 

whence, NM'^ = (r — 40) (r -4- 40). 

Havinsr determined NM, subtract VL from it, and we have 
Nt, one of the ordinates required. 



BOOK IX.] 



RAILWAY CURVES. 



329 



In this manner, by calculating the full ordinate to the 
diameter, and subtracting FZ, any desired number of offsets are 
determined for the half chain VF. For FWy the ordinates have 
the same length, but are located in the inverse order. 

The middle ordinate, FE, is found by subtracting VL from 
the radius. 

E XA M PLE. 

Determine the ordinates 10 feet apart on a 100-foot chord, 
for a two-degree curve. Eadius, 2864.79 feet. 

Ans. At 10 feet = .15 feet. 

At 20 " = .28 « 

At 30 " = .36 " 

At 40 " = .42 " 

Middle ordinate = .43 " 

Reversed and Compound Curves.* 
362. Two curves, of the same or different radii, may join 
each other and have a common tangent at the point of junction 
If the curves lie on opposite sides of the common tangent, they 
form a reversed curve, and their radii may be equal or unequal ; 
if they lie on the same side of the common tangent, they have 
unequal radii and form a compound curve. Thus ABC (Fig. 174) 
is a reversed curve, and 
ABD a compound curve. 
The point of contact 
of the common tangent 
is called, in a reversed 
curve, the reversing point, 
and, in a compound 
curve, the common tan- 
gent point or the point 
of compound curvature. Fiq. m. 

* Taken, by permission and with slight alterations, from Henck's " Field Book for 
Railroad Engineers,'" 




^30 



ELEMENTS OF SURVEYING. 



[book IX. 




Fig 175. 



The reversing point of a reversed curve contained be- 
tween parallel tan- 
gents is in the line ^ ^ 
joining the tangent 
points. Thus, in the 
curve ACB, Fig. 175, 
contained between the 
parallel tangents HA 
and BKj the reversing 
point, G, is in the line 
AB joining the points of contact A and B, 

A reversed curve is of use in certain track-work near stations, 
constructing turnouts, &c., though it is not, or ought not to be, 
used on the ordinary running track. 

363. Given the perjjendicular distance letween two parallel 
tangents BD = b (Fig. 175), the distance hetiveen the two tangent 
points AB = a, and the first radius, EG ^ R, of a reversed 
curve uniting the tangents HA and BK, to find the chords 
AG ^= a' and GB = a", and the second radius GF = R', 

Draw the perpendiculars EG and FL (Fig. 1 75). Then the 
right-angled triangles ABD and EAG have the angle BAD 
= AEG, since each is one-half AEG, and are, therefore, 
similar ; hence. 



AB 



or 



Since 



a 



BD 

: h 



a 



EA : AG, 
R : ^a\ 
2Rb 



a 



a" -\- a' = a, 



a = a 



a . 



(1) 



(2) 



To find R' , we have, from the similar triangles ABD and FBL, 

aih : : R' : \a". 



BOOK IX.] 



RAILWAY CURVES. 



331 



„, aa 



(3) 



Any three of the quantities a, a', a", b, R, R, being given, the 
others may be found from equations (1), (2), (3). 

EXAMPLES. 

1. 5 = 8, « = 160, i? = 900 ; find a', a", R'. 

a' = 90 ; a" = 70 ; R' — 700. 

2. ^ = 8, a' = 90, a" =z 70 ; find a, R, R', 

d. R=: 900, R' = 700, ^^ z= 8 ; find «, a', a", 

364. Given the line AB := a (Fig. 176), ivJiich joins the fixed 
tangent points A and B, the angle DAB = A, and the angle 
ABG =z B, to find the common radius EG=GF=:R of a 
reversed curve to unite the tangents HA and BL. 

From the triangle 
ABK, Fig. 176, we 
have (Art. 32), 

AE . EK : : sin AKE 
: sin EAK', 

but since 

EAK= 90° -^ 

sin EAK = cos ^; 

hence, denoting angle AKE by K, we have, 

EK -sin K= R- cos A, (1) 

In like manner, from the triangle BFK, we have, 

FK'Sm K= R ' cos B. (2) 

From (1) and (2), by addition, we have, 

(EK+FK) sin K = 2R sin K ==: R (cos A +cos B). (3) 



H 




Fig. 176. 



332 ELEMEITTS OF SURVEYING. . [BOOK IX. 

. • . sin ^ = 1^ (cos A + COS B). (4) 

But cos ^4-cos ^ = 2 cos J (A-{-B) cos J (A—B), 
(see Legendre, Trig. Art 67) ; hence, 

sin X= cosi{A-\- B) cos ^ (A—B). (5) 

Having found K, we have the angle 
AEK= E — 180°-(^+^^^) = 180°-(^+90°-^) 

:=90°-f^-^. (6) 

In like manner, the angle 

BFK = F- 90° + B—K. (7) 

From the triangle AEK, we have 

AE : AX : : sin ^ : sin E, 
.-. i?sin ^=1 ^^sin JT. (8) 

In like manner, from triangle BFK, we have, 

i? sin i^ = BK sin K (9) 

From (8) and (9), by addition, we have 

R (sin ^+sin F) = (AK+BK) s'm K =z a sin K. (10) 
But, sin ^+sin i^ = 2 sin J (E+F) cos ^ (E—F), 
(see Legendre, Trig. Art, 67) ; hence, 

2R sin i (E+F) cos J (E—F) = a sin K. 

. D_ i^ sin X 

'• sin^(^+i^)cosi('^-i^)* ^ ^ 

Substituting in (11) the values of ^ and F from (6) and (7), 

we have 

p ^a^n_K , . 

~ cos [^-i (^ + ^)] COS i {A-B) ^^^ 

Example.— Given a = 1500, A = 18°, ^ = 6° ; find i?. 
From equation (4), K is found to be 76° 36' 10", and then 
from (12), R is found to be 1710.48. 



BOOK IX. J 



RAILWAY CUEVES. 



333 



365. If one branch of a compound curve he produced until 
the tangent at its extremity is parallel to the tangent at the 
extremity of the second branch, the common tangent point of the 
two arcs is in the straight line produced which passes through 
the tangent points of these parallel tangents. 

Let ACB, Fig. 177, ^ 

be a compound curve 
uniting the tangents HA 
and BK, The radii UG 
and FC, being perpen- 
dicular to the common 
tangent at C, the point 
of compound curvature, 
are in the same straight 

line. Continue the curve p 

AC to D, where its tan- fig. itt. 

gent OD becomes parallel to BK, and consequently the radius 
ED parallel to FB, Then if the chords CD and CB be 
drawn, we have the angle CFD = CFB, whence BCD, the half- 
supplement of CFD, is equal to FCB, the half-supplement of 
CFB. But FCD can not be equal to FCB, unless CD coincides 
with CB; therefore, the line BD produced passes through the 
common tangent point (7. 




366. To find a limit in one direction of each radius of a 
compound curve. 

Let AI and BI, Fig. 177, be the tangents of the curve ; draw 
IM bisecting the angle AIB ; draw ^X and ^il/ perpendicular 
respectively to AI and BI, meeting IM in L and M. Then the 
radius of the branch commencing on the shorter tangent, A I, 
must be less than AL, and the radius of the branch commencing 
on the longer tangent, BI, must be greater than BM. 



334 ELEMENTS OF SURVEYING. [bOOK IX. 

For, suppose the shorter radius equal to AL, and hence, /iV' 
equal to Al; join LJV, then the equal triaugles AIL and iVTZ 
give AL = LN', so that the curve, if continued, will pass 
through N, Avhere its tangent will coincide with IN. Then 
(Art. 365) the common tangent point would be the intersection 
of the straight line through B and N with the first curve ; but 
in this case there can be no intersection, and therefore no com- 
mon tangent point. 

Suppose, next, that this shorter radius is greater than AL, 
and continue the curve till its tangent becomes parallel to BI. 
In this case, the extremity of the curve will fall outside the tan- 
gent BI in the line AN produced, and a straight line through 
B and this extremity will again fail to intersect the curve already 
drawn. As no common tangent point can be found when the 
shorter radius is taken equal to AL, or greater than AL, no 
compound curve is possible. This radius must, therefore, be less 
than AL. 

In like manner it may be shown that the radius of the other 
branch of the curve must be greater than BM. 

If the tangents AI and BI, and the intersection angle / are 

known, then . 

AL =: AI tan J J; 

BM = BI tan ^L 

These values are, therefore, the limits of the radii in one 
direction. 

367. If nothing were given but the position of the tangents 
and the tangent points, it is evident that an indefinite number 
of different compound curves might connect the tangent points ; 
for the shorter radius might be taken of any length less than the 
limit found above, and a corresponding value for the greater 
could be found. Some other condition must, therefore, be intro- 
duced, as, for example, in the following problem: 



BOOK IX.] RAILWAY CURVES. 335 

368. Given the line AB = a (Fig. 177), which joins the fixed 
tangent poijits A and B, the angle BAI ^= A, the angle ABI=z B, 
and the first radius AE = E, tofiiid the second radius BF= R' 
of a compound curve to unite the tangents HA and BK. 

Let the first curve be ruQ with the given radius from A to D, 
where its taugent DO becomes parallel to BI ; then the common 
tangent point G is in the line BD produced, and the chord 
GB= GD^BD. 

The angle GAD, formed by a tangent and a chord, is 
measured by half the arc AGD ; hence, R sin GAD = R sin 
IAD =z ^AD (see Legendre, Trig. Arts. 64 and 30) ; hence, 

AD = 2R sin IAD. (1) 

In the triangle GAD, since GA and GD are equal, angle AGD = 
1SO°—20AD; hence, GAD=IAD=90°—iAGD=90°—iAIBj 
but from the triangle AIB, ^AIB = 90°—^ {A-}-B); 

whence the angle IAD = ^(A-\-B), (2) 

From (1) and (2), we have 

AD =:^2R sin i (A -\-B). (3) 

Then in the triangle ABD, we have AB = a, AD =: 2R sin 
J (A-\-B), and the included angle 

DAB = lAB-IAD = A-i {A-{-B) = i (A-B) ; 

whence, we have the proportion (Legendre, Trig. Art. 45), 

AB-^AD : AB—AD :: teiu i(ADB-\-ABD) 

: tsini(ADB-ABD). (4) 

i {ADB + ABD) = i {180°— DAB) ; i {ADB—ABD) may be 
found from (4); and from the half-sum and half-difference thus 
obtained, the angles ADB and ABD may be found. These 



336 ELEMENTS OF SURVEYING. [bOOK IX. 

angles being known the side BD may be found from the 
proportion (Art. 32), 

sin ABD : sin DAB :: AD : BD. 

Tlie angle CBI = B—ABD. 

Again, as above, CB = 2i^' sin OBI, 

and CD = 2R sin CDO = 2E sin CBI. 

Substituting these values of CB and CD, in the equation 

CB= CD + BD, 
we have 2R' sin CBI = 2E sin CBI-^BD. 

HR'-R)^^-c-B-i- («) 

When the angle B is greater than A, that is, when the greater 
radius is giyen, the solution is the same except that the angle 
DAB =z ^[B—A), and CBI is found by subtracting the supple- 
ment of ABD from B. We shall also find CB = CD—BD; hence, 

^'^^-2-si^^' (^^ 

or 2 (R-E) = -^^. 

^ ' sm C7^/ 

Note. — If more convenient, the point D may be determined 
in the field by laying off the angle IAD — ^ {A -\- B), and 
measuring the distance AD ■= 2R sin ^ (A-\-B). BD and CBI 
may then be measured, instead of being calculated as above. 

Example, a = 950, A = 8% B = r, R = 3000 ; find R'. 
AD = 2x 3000 sin ^ (8° + 7°) = 783.16 



BOOK IX.] KAILWAY CURVES. 337 

and DAB = ^ (8°-7°) = SO'. 

Then, to find ABD, we have 

log (AB—AD), 166.84 = 2.222300 

log tan i (ADB + ABD), 89° 45' = 2.360180 

(a. c.) log {AB-^AD), 1733.16 = 6.7611 61 

log tan i {ADB—ABD), 87° 24' 17" = 1.343641 
.-. ^^Z) = 2°20'43"; 
whence, CBI = B—ABD = 4° 39' 17". 

Next, to find BD, 

log AD (783.16) = 2.893849 

log sin DAB (30') = 7.940842 

(a. c.) log sin ABD (2° 20' 43") = 1.3880 52 

log BD (167.01) = 2.2227 43 

Then, from (6), we have 

log BD (167.01) = 2.222743 

(a. c.) log CBI (4° 39' 17") = 1.0907 08 

log 2 (B'-B)y 2058.03 = 3.3 13451 

.-. R'-R = 102dM; 

R' = 3000 + 1029.01 = 4029.01. 



BOOK X. 

MINING SURVEYING. 



SECTION I. 

DEFINITIONS AND GENERAL PRINCIPLES." 

369. Mining Surveying comprises all the operations neces- 
sary to determine the relatiye positions of the parts of a mine 
with respect to each other, and also with respect to the surface 
of the earth. 

370. The general principles involved in this branch of sur- 
veying are the same as those used in surface surveying, but, 
from the nature of the case, certain modifications are required. 

Stations are designated by lamps instead of flags, and lamp- 
stands instead of flag-rods, or by plumb-lines properly suspended 
and lighted ; station points, if temporary, are marked by cross- 
lines chipped in the rock, or sometimes by simple chalk lines, 
and, if permanent, by iron pegs driven into holes drilled for the 
purpose. 

The compass is rarely used for underground work, and 
ought never to be used for any but rough work, because of the 
inaccuracies to which it may lead. A great deal of the mining 
in the U. S. is done in the far West, where the magnetic declina- 
tion is not known ; and, further, the declination is seriously 
affected by the state of the atmosphere, the presence of iron ores, 
magnetic pyrites, &c. 

The transit, which is the principal angular instrument em- 




SEC. I.] DEFINITIONS. 339 

ployed, and the only necessary one, differs from the ordinary 
transit in having a diagonal eye-piece, to permit observations to 
be made when the telescope is directed vertically upward, and 
also an arrangement for illuminating the cross-wires. 

A method of lighting the cross-wires is by the reflector shown 
in Fig. 178. This is an elliptical piece of brass, 
silver-plated on the under side, and inclined at 
an angle of 45° to its ring, which is fitted to the 
object end of the telescope ; the hole in the 
reflector admits the use of the telescope, while 
a light held near the under surface illuminates ^^^' i'*^- 

the cross-wires. 

The transit should be furnished with a solar attachment (see 
Appendix A) for establishing the true meridian and running out 
lines with reference thereto ; and should also have an extension 
tripod to adapt it for use in mountain surveys, where one or 
more legs must be shortened, and for mines, where in many 
places a short tripod is indispensable. 

371. Traversing is the operation of running a zigzag line, 
from one point to another. The elements of the traverse are 
straight lines, determined by their lengths and by their in- 
clinations to certain fixed planes. In mining surveying, three 
such planes are used ; the first, is either a meridian plane 
through the origin of the traverse, or a vertical plane through 
the first course; the second, is a horizontal plane through the 
origin ; and the third, is a vertical plane through the origin, 
and perpendicular to the other two. 

372. "Working, or Reducing the Traverse, is the opera- 
tion of finding the length and direction of a single line, 
equivalent to the zigzag, that is, starting from, and terminating 
at, the same points. Such a line is called the resultant of the 
traverse. 



340 



ELEMENTS OF SURVEYING. 



[book X. 



The zigzag line is run along the subterranean openings of 
a mine. Such openings, when vertical, are called shafts, 
and when not vertical, tunnels. 



SECTION II. 



METHOD OF LOCATING CLAIMS. 



373. The methods employed in locating claims on the surface 
are various, and in general very crude, except when a surveyor, 
usually a U. S. Deputy Mineral Surveyor, is called upon to make 
the survey. Prospectors are usually without suitable instruments 
to lay off their claims with any degree of accuracy. We will. 






-%^ 



S5^^" 



^, 



3l 



^. 



,,--^- 



is^ 







^■82'"-^... 



20^^ 






T 9 XR 5T\^ 



Second Standurd. 
Fis. 179. 



TaralUl 'S 



% 






SEC. II.] METHOD OF LOCATING CLAIMS. 341 

therefore, only consider the method employed by the Deputy 
Mineral Surveyors, who are compelled to follow the instructions 
received from the Land-office. 

374. The mining claim shown in Fig. 179 has the dimensions 
allowed by the U. S. Mineral Laws, viz., 1500 feet in the direc- 
tion, or on the ^^ strike," of the vein, by 300 feet on each side of 
the middle of the vein at the surface. An ideal case is assumed, 
the exact strike is supposed to have been determined, and the 
side lines of the claim run parallel to it. Such cases are the 
exception. Generally, the strike is not accurately determined 
by the prospector, and claims are, in both direction and dimen- 
sions, usually but a rough approximation. The courts are lenient 
and allow considerable latitude in the matter. The prospector 
may alter the boundaries of, or *' swing," his claim after the strike 
has been determined, provided that no trespass is committed ; 
but when the course cannot be altered without interference 
with other claims, the boundaries already established must be 
adhered to. 

When a claim is made and staked out, a record of the same 
must be placed on file in the office of the Eecorder having charge 
of the records of mining locations in the district where the claim 
is situate ; and in case the boundaries of such claim are altered, 
a record of the amended location must be filed in like 
manner. 

*^A failure to make and record the location in accordance 
with the law and regulation in force at the date of the location, 
will defeat the claim ; and if it is not made with such definite- 
ness as to operate as notice to all persons seeking to acquire 
rights to mining lands, it will be void for uncertainty " (General 
Land Office Instructions). 

375. When mineral discoveries are made and located on sur- 
veyed land, the surveys must conform to the public survey and 



342 ELEMENTS OF SURVEYIl^G. [BOOK X. 

be connected with or ^^tied" to it, so that there will be no diffi- 
culty in finding some established corner. In Fig. 179, the points 
1 and 2 are tied to the N. W. cor. of the quarter-section in 
T. 9 N. R. 5 W. 

376. When discoveries are made on unsurvejed land, U. S. 
locating monuments are established to which the claims are 
tied in the same manner. These monuments may be natural 
objects, as mountain peaks, permanent rocks, or they may be 
artificially constructed, in which case they should be strong and 
without liability to disturbance. The method of tying to such a 
monument is shown in Fig. 179. When a corner is properly 
tied in the manner mentioned, the courses are staked out, 
1500 ft. along the length of the vein by 300 ft. on each side of 
the same. The end lines must be parallel to each other in 
accordance with the Mining Law, but the side lines need Dot be 
so, though the survey of a lode ^'must be substantially a 
parallelogram." 

377. Before commencing the survey for a patent, the original 
stakes are found, if possible ; that is, if no amended location cer- 
tificate has been filed. In the latter case, the stakes marking 
the amended location are the ones to be followed. When no 
stakes can be found, the surveyor will be guided by the locator or 
some one familiar with the boundaries of the property, provided 
the boundaries then pointed out do not differ from the record of 
the claim on file in the Recorder's office. Even in the latter 
case the surveyor can, at the wish of the owner, and when there 
is no interference with other locations, alter the claim (swing it), 
file an amended location notice in the Recorder's office, and send 
forward the description and maps of the altered claim for 
patent. 

378. In making a survey, it is frequently found that there 
are conflicting claims, that is, two or more claims cross or lap 



SEC. II.] 



METHOD OF LOCATING CLAIMS. 



343 



over each other. An example is shown in Fig. 180, which is a 
plot made from actual claims on Treasure Hill, Black Hills, 




Fig. 180. 



Dakota. When claims conflict, priority of location must govern 
as to the ownership of the surface in dispute, and the U. S. 
Deputy Surveyor in making his plot, etc., is required, by the 
land-office instructions, to show and deduct the area in conflict 
from the subsequent location, — as is shown in the figure. Should 
this not be satisfactory to the owner, his appeal is to the 
courts. 



344 ELEMENTS OE SURVEYIN'G. [BOOK X. 

379. When the survey for patent is completed, the surveyor 
drives strong stakes, generally about 4 inches to 5 inches square, 
and 4 to 5 feet long, firmly into the ground, and marks them 
M. S. (Mineral Survey) No. 1, M. S. No. 2, etc. No less than 
4 stakes, one in each corner, can be used to mark a claim, and in 
some cases more are needed. In Fig. 180, for example, it is 
readily seen that four stakes would not be sufficient to trace out 
the boundaries of the Golden Seal, Ocean Wave, Elkhorn, or 
Fenian claims. 

380. When claims are surveyed for patent, they are numbered 
in the order of their application. Lot No. 1, No. 2, No. 3, etc. 
Figure 180 shows that but 3 of the claims represented were 
patented at the time the survey was made, viz.. Placer Lot No. 
120 and 55, and Esmeralda Lot No. 226. The Esmeralda being 
patented, it will be seen that the areas of the other claims in 
conflict with it have been deducted, so also of the others in the 
order of their priority. 

381. With regard to errors and corrections in field and office 
work, in a survey for patent, the following instructions are given 
to Deputies (see Special Instructions to Deputies by the Survey- 
ors-General of Montana and New Mexico) : 

" The error in balancing must not exceed two feet in one 
thousand feet, and this will be allowed only on complicated sur- 
veys or when the ground is very rough." 

In the notes of survey returned, must be a table of area cal- 
culated by double meridian distances; in this table must be 
shown first the balance of the survey as actually run upon the 
ground, and immediately thereafter the 'corrected latitudes and 
departures. The correction must be made, either by distributing 
the eiTor among the courses, in proportion to the length of each 
course, by the proportion — The sum of the lengths of aU the 
courses is to the total error in latitude or departure as the length 



SEC. II.] METHOD OF LOCATING CLAIMS. 345 

of each course is to the correction of its latitude or departure 
(see Art. 127) ; or by the following method, which is considered 
preferable : 

*^ You can correct the balance by computing a closing course 
and distance. This can be made on the last line of the survey 
when it is one of the longest lines, but otherwise should be made 
upon the longest line unless there should be a line that could not 
be run accurately, and in that case close on it, when you use this 
method. This corrected line will be the true line to be used in 
the notes and on plot. In notes you will state immediately fol- 
lowing (parenthetically) the course and distance of the line as 
actually run upon the ground. This is the better way, as the 
actual error made in the survey can be detected by reference to 
this line only." 

" The maximum error that will be allowed in a lode claim is 
0.03 acres." 

382. The instructions of the U. S. General Land Office to 
U. S. Deputy Mineral Surveyors are embraced in a series of 
decisions and letters to the U. S. Surveyors-General, relating to 
particular cases. Based upon these decisions and letters, and 
upon what in their judgment and experience seemed necessary, 
the Surveyors-General have generally issued printed instructions 
to their deputies. 

A copy of the instructions of the Surveyor-General of 
Colorado to Deputy Mineral Surveyors, with sample Field 
Notes and sample Plot furnished therewith, is given in Ap- 
pendix 0. 



346 ELEMENTS OF SURVEYING. [BOOK X. 

SECTION III. 

UNDERGROUND TRAVERSING, ETC. 

383. Let it be required to sink a shaft from the surface to 
connect with a tunnel, which is driven into a hillside. 

In mauy cases the needle cannot be used on account of 
disturbing influences, such as the proximity of iron and its ores, 
or magnetic pyrites. The vernier readings of the transit are then 
employed, and indeed this method is preferable in all cases. 

Ascertain the course of the tunnel and the position of the 
point where it is desired to make the connection. Establish a 
station -stake at any convenient distance from the mouth of the 
tunnel that will command a sight of its centre point. If the 
centre line of the tunnel is a straight line, or nearly so, a sight 
may be taken from the established station to the point of inter- 
section. In general, however, this cannot be done, the line of 
sight being cut off by deflections in the course, thus requiring a 
traverse. 

384. Where curves occur, stations should be established by 
driving nails in the centre of the cap-timber, overhead in the 
tunnel. If there are no timbers, holes should be drilled in the 
rock and filled with wooden spuds into which nails are driven. 
Plumb-lines are then suspended from these nails, lighted by 
lamps suitably placed, and used for sighting to with the 
transit. 

If desired, the selected stations may be marked by iron pegs 
driven into holes drilled in the floor of the tunnel. In this case, 
guiding lamps placed on stands similar to tripods, but with 
sliding pieces carrying the lamps and fixed in position by clamp- 
screws^ are used for sighting to. By the sliding and clamp 



SEC. III.] UKDERGROUKD TRAYERSIN^G. 347 

arrangement, the height of the lamp may be made equal to that 
of the line of sight of the transit telescope. 

As the same methods will apply in either case, the suspended 
plumb-lines will be assumed as giving, in general, better 
results. 

385. To place the transit accurately in position at a station, 
take a block of lead with a steel point in the centre projecting out 
about half an inch ; place it on the floor of the tunnel, with the 
steel point upward, immediately under a plumb-bob let down 
from the station overhead ; the plumb-line may then be removed 
and the transit set in position over the steel point. 

386. Figure 181 shows a somewhat extreme case, chosen to 
illustrate the method of traversing. Take a station at A, out- 
side and 100 feet (say) from the mouth of the tunnel. Fix a 
stake at Ay and drive a nail in the top of it for convenience and 
accuracy of sighting. Calling the outside station No. 1, and the 
one at the mouth of the tunnel No. 2, take the line between 
stations 1 and 2 as the meridian of the survey, find the azimuths 
of the several successive courses, 2 to 3, 3 to 4, &c., with respect 
to this meridian, precisely as directed in Art. 194. The bearing 
of the several courses with respect to this assumed meridian may 
then be found and tabulated as directed in Art. 195. 

If necessary, measurements are made to determine the cross- 
section of the tunnel, its height or breadth, at a station, or at any 
desired point of the course, and the results entered in the column 
of remarks in the field-book. 

387. The distances between the stations are now measured 
with great care. A chain should not be used in these measure- 
ments ; a light steel tape is much better. The points corres- 
ponding to the ends of the tape are marked by chalk lines on the 
rock, or in some other convenient manner. The distances may 



348 



ELEMEin^S OF SUBVEYrN'G. 



[book X. 



be measured horizontally, or along the slope Of the tunnel, in 
which latter case they must be reduced to horizontal distances. 



OVi3ruBnj^ ^ 




388. When distances between stations are measured along the 
slope, the angle of elevation or depression of the course must be 
determined — and may be determined as follows: 



SEC. III.] 



UNDERGROU"N^D TRAVERSING. 



34^ 



Let the nail on staff at A (Fig. 181) be at the same distance 
above the ground at A that the axis of the vertical limb of the 
transit is above the floor of the tunnel at station 2 ; when the 
telescope is pointed to the nail at A, the reading of the vertical 
limb will be the angle of inclination of the first course, in eleva- 
tion, if the telescope points downward — in depressio7i, if it points 
upward. Let the point of a plumb-bob, suspended from the 
nail overhead at station 3, at the same distance above the floor of 
the tunnel at 3 that the axis of the vertical limb of the transit is 
above the floor at 2 ; when the telescope is directed to the point 
of the plumb-bob, the reading of the vertical limb will be the 
angle of inclination of the second course, in elevation, if the 
telescope points upward — in depression, if it points downward. 
In like manner are obtained the angles of elevation or depression 
of the several successive courses to the end of the traverse. 

389. The method of recording the observations and measure- 
ments made in a traverse, is shown in the following 

Field Book. 



stations. 


Angles of Inclination. 


Azimuth with 
Course 1.2. 


Distance 
in feet. 


Remabks. 


Elevation. 


Depression. 


1 

2 
3 

4 


0° 0' 


0° 0' 


0° 0' 
0° 0' 
340° 
300° 


102 

100 
150 
180 


Sta. 1, at staff, outside 
tunnel. 

Sta. 2, at iron peg, at 

mouth of tunnel 

overhead. 


5 






280° 


200 


At Sta. 5, breadth 6 J ft. 


6 






325° 


120 




7 






275° 


162 


At sta. 7, height 7J ft. 


8 






265° 


120 


At iron peg. 


9 













350 



ELEMENTS OF SURVEYING. 



[book X. 



The above is a record of the traverse shown in Fig. 181. 
The distances measured are supposed to be horizontal distances, 
and the angles of inclination are, therefore, zero. 

390. The method of reducing the traverse is as follows . 

From the azimuths of the several courses with the given 
course taken as meridian, find the bearings (see Art. 195) 
and proceed as shown in the following 



Office Form. 



o 


Slope in Feet. 


Bearing with 
Course 1.2. 


Length 

of 
Course. 


Latitude. 


Departuke. 


Eleva- 
tion. 


Depres- 
sion. 


N. 


s. 


E. 


w. 


1 








North 


102 


102 








2 






North 


100 


100 








3 






N20° W 


150 


140.95 






51.30 


4 






N60° W 


180 


90 






155.88 


5 






N80° W 


200 


34.72 






196.96 


6 






N35" W 


120 


98.30 






68.83 


7 






N 85° W 


162 


14.12 






161.38 


8 
9 






S 85° W 


120 




10.46 




119.54 


fl 










580.09 


10.46 


00 


753.89 


es 

OQ 






N52°56'W 


944.68 


10.46 






00 


569.63 


753.89 



Hence the resultant course A G, from 1 to 9, has a northing 
of 569.63 feet and a westing of 753.89; to find its length and 
bearing, we have in the triangle ABC, 



hence 



AB = 753.89 and BC = 569.63 ; 
AC = VaB^ ^'M^ = 944.89. 



SEC. III.] 



UNDERGROUND TRAVERSING. 



351 



tmBAC 



BC 
AB 



569.6 3 
753.89 



BAG= 37° 4'; 

.-. bearing oi AG = 90° - 37° 4' = 52° 56'. 

391. If the distances between stations are measured on the 
slope, instead of horizontally, the angle of elevation or depres- 
sion of each course must be observed, and recorded in the proper 
column of the field book. The distances on the slope must 
then be reduced to horizontal distances in finding the resultant 
course. The method of record and reduction in such a case, 
and when the magnetic bearing of the course, from which 
azimuths are measured, is given, will be apparent from the 
following 

Example. Eequired the resultant of the traverse of which 
the notes are contained in the following 

Field Book. 



stations. 



Angles op Inclination. 



Elevation. 



1°30' 

2° 00' 



Depression. 



2° 30' 
3° 15' 
3° 30' 

4° 15' 



Azimuth with 
Course 1.2. 


Distance 
in feet. 


0° 0' 


307 


176° 15' 


402 


228° 30' 


240 


297° 


367 


246° 15' 


409 


249° 45' 


200 



Remarks. 



Sta. 1, at iron peg, 
centre of shaft. 



At iron peg. 



Let the magnetic bearing of the course, 1.2, from which 
azimuths are measured, be S 23° W. 



fla 



352 



ELEMEl^TS OF SURVEYII^a. 



[book X. 



Find, from the bearing of the course taken as meridian and 
the azimuths of the several courses, the bearings of the several 
courses, as in Art. 196. The traverse is then reduced as in the 
following 

Office Form. 





Slope in "Feet. 




Reduced 
length 

of 
course. 


Latitttde. 


Depabttjre. 


Course. 


Eleva- 
tion. 


Depres- 
sion. 


Bearing. 


N. 


s. 


E. 


\V 


1 




13.4 


S 23° W 


306.7 




282.3 




119.8 


2 




22.8 


S 191° W 


401.4 




378.9 




132.3 


3 




14.7 


S 71i° W 


239.6 




76.0 




227.2 


4 




27.2 


N40° W 


366. 


280.4 






235.3 


5 


10.7 




S 89^° W 


408.9 




5.4 




408.9 


6 


7. 




N87i°W 


199.9 


9.6 






199.7 




17.7 


78.1 

17.7 






290. 


742.6 
290. 




1323.2 
0.0 


Result- 
ant. 




60.4 


S71°7'W 


1406.9 




452.6 




1323.2 



The length of the course on the slope, multiplied by the sine 
of the angle of inclination, gives the distance the course rises or 
falls, in feet (if the measurements are made in feet) ; and the 
length multiplied by the cosine of the angle of inclination, gives 
the reduced length of the course, that is, the length that would 
have been found had the course been measured on a horizontal 
line. The bearing and reduced length of each course being 
found, the latitudes and departures of the several courses are 
found as before. In this example, the resultant course descends 
60.4 feet, its southing is 452.6 feet, and its westing 1323.2; its 
bearing, S. 71° 7' W., and length on the horizontal, 1406.98, are 
then found as before. 

392. The difference of level of two points underground may 



SEC. III.] 



UNDERGR0U:N^D TRAVERSIl^a. 



353 



be found as shown in the example given in Art. 391, or by 
running a line of levels between the points as directed in 
Art. 294. Leveling rods for underground use should be 
painted and graduated to feet, tenths, and hundredths of a 
foot, and about 3|- to 5 feet in length. When reading the 
graduations of the rod through a telescope, the flame of two 
lamps, one from each side, should be brought to bear on its face 
so as to light it and not obstruct the line of sight of the observer. 



Method of Plotting the Underground Traverse on the Surface. 

393. To plot the traverse on the surface of the earth, the 
first course, from which azimuths were measured, must first be 
laid down, in direction and horizontal length, and its two ends 
marked with suitable pegs. If the first station has been taken 
outside the tunnel, as in the traverse shown in Fig. 181, the first 
course is already, or may easily be, pegged out on the surface. 
If, however, the first course is within the tunnel, as in example 
given in Art. 391, the first course must be connected with thf 
surface. There are two principal methods of making the con- 
nection : 

394. FIRST METHOD. 

A straight-edge, AB^ 
is mounted on two 
trestles, and from it are 
suspended two plumb 
lines, E and F, as far 
apart as the breadth of 
the shaft will permit. 
To prevent agitation 
from currents of air, the . 
bobs are permitted to dip 
into buckets of water, at Pio, igg. 




354 



ELEMENTS OF SURVEYING. 



[book X. 



the bottom of the shaft ; the transit being at the second station, 
K, and the telescope turned in the direction of the first station, 
D, the straight-edge is moved by an assistant until both are seen 
in line from K ', their plane then passes through the first course; 
and if the line ^^ be prolonged to M and L, the line ML will be 
directly over the first course, and consequently its bearing will be 
that of the first course. By measuring the line E^ the depth of 
the shaft may be found. 




Fig. 183. 



395. SECOND METHOD. 

Let the transit, provided 
with a diagonal eye-piece, 
be planted over the station 
D, at the foot of the shaft, 
and after being leveled, let 
it be directed on the sta- 
tion K. Then, without 
changing the plane of 
vision, let the transit be 
directed to the top of the 
shaft, and let an assistant plant two flag rods, one at A and the 
other at B, both in the plane of vision, and let the line AB be 
prolonged to L and M, as before. The line XJ/ will be in the 
same vertical plane with the first course, DK. Hence, as before, 
we may determine the bearing of the first course of the traverse. 

396. Having, on the surface, the direction of the first course 
and the position of Station 1 of the traverse. Station 2 of the 
traverse may be found by laying off from Station 1, in the proper 
direction, a horizontal distance equal to the length of the first 
course. The position of Station 2 on the surface will thus be 
determined. Put the transit over the surface position of Station 
2, make the zero of the vernier coincide with the zero of the 



SEC. III.] U2fDERGR0UND TRAVERSIKQ. 355 

horizontal limb and clamp the vernier plate ; direct the telescope 
to surface Station 1 and clamp the limb ; revolve the telescope on 
its horizontal axis, unclamp the vernier plate and revolve it 
around to the right, through the azimuth of the course, from 2 
to 3 ; the line of sight of the telescope will then be in the direc- 
tion of the traverse line from Station 2 to Station 3 ; measure 
from surface Station 2, in the direction found, a horizontal 
distance equal to the horizontal distance of the traverse line 
from Station 2 to Station 3 ; the extremity will be the surface 
position of Station 3. Clamp the vernier plate and remove the 
transit to surface Station 3; reverse the telescope on its hori- 
zontal axis, loosen the lower clamp and sight surface Station 2 ; 
tighten the lower clamp and revolve the telescope on its axis ; 
unclamp the vernier plate and turn it through the azimuth of 
the course from 3 to 4 ; the line of sight of the telescope will 
then lie in the direction of the traverse line from Station 3 to 
Station 4 ; determine the surface position of Station 4 by 
measurement from surface Station 3, as before (see Art. 196). 
Proceed, in like manner, to mark out on the surface the suc- 
cessive underground stations and courses, till the required point 
(9 in Fig. 181) is found. 

397. The slope rod is a convenient instrument for use in 
making surface measurements. 

A device for such a rod, for which caveat was entered by A. J. 
Rigby of New York in 1883, consists essentially, as shown in 
Fig. 184, of a horizontal bar 10 to 12 feet long, made to move up 
and down a vertical bar 6 to 8 feet long, to which it may be 
clamped in any position. 

The horizontal bar. A, carries a scale, graduated to hun- 
dredths of a foot and reading by vernier to thousandths of a foot, 
a spirit level E, by which the bar may be made truly horizontal, 
and an extension bar C, which may be drawn out 4 to 6 feet, 



356 



ELEMENTS OF SUKYETING. 



[book X. 



making the extended bar 14 to 18 feet long ; when the bar is drawn 
out, the reading is made by a vernier on the extension bar, as 
shown in the figure, in a manner similar to that in which the 



^ SorixantdZ Bar 
S TerticdlBar. 
C Esiension^ JBar. 
X> Clamp Plabe. 
E SpiritXievel. 
F Clcanp Screws. 



r 



^^■. 



9 22'] 

c 



v. 




Fig. 184, 

extended leveling rod is read (see Art. 290). The vertical bar, 
B, also carries a graduated scale, reading by vernier to thou- 
sandths of a foot. The two clamp-screws, FF, working against 
a clamp-plate, fasten the horizontal to the vertical bar and keep 
it firmly in position, both horizontally and vertically, against its 
bearings on the vertical bar. 

The manner of using the slope-rod is shown in Fig. 185. 
Pegs are driven into the ground at convenient stations along the 
line which is to be measured, the exact point on each being 
marked by the intersection of cross-lines on the top of the peg ; 
if the selected station is on a rock, and a peg can not be driven, 
the cross-lines are marked in the rock to note the station. The 
front edge of the vertical bar is placed at the point marked on 
one peg and the end of the horizontal bar, extended if necessary, 
to the point marked on the other ; the horizontal bar is moved 



SEC. ni.] 



UNDERGROUND TRAVERSING. 



357 



slowly up and down the vertical bar till the bubble of the spirit- 
level E is in the middle of its run, and then clamped in position. 




Fig. 185. 



The reading on the horizontal bar will give the true horizontal 
distance between the two stations, and the reading on the 
vertical bar will give their difference of level. 

398. A better method than repeating the traverse on the 
surface will appear from a consideration of the diagram, Eig. 181. 
It will be seen that the algebraic sums of the latitudes and 
departures respectively of the several courses, with respect to 
the first course taken as a meridian, are equal to the latitude and 
departure of the last or station 9. The latitude of 9 is the perpen- 
dicular of a right-angled triangle and the departure is the 
base ; the distance A C, from 1 to 9, is equal to the square root 
of the sum of the squares of the latitude and departure of 9. 
The angle BA C may be found by the relation 

tan A = -^-j (Art. 35). 

Having found the angle BAC, the bearing of AC, with 
respect to the course 1.2 taken as meridian, is equal to 
90° — BAC. Place the transit over station 1, level it, make the 
zero of the vernier coincide with the zero of the horizontal limb. 



358 



ELEME>sTS OF SURVEYII^G. 



[book X. 




I 



SEC. III.] CKDEKGEOUND TRAVEKSING. 359 

clamp the vernier plate, sight to station 2 and clamp the limb ; 
unclamp the vernier plate, turn it through the bearing, as found, 
of A Cy and clamp it ; the line of sight of the telescope will be in 
the line A C. Measure the horizontal distance, as found, from A 
to Cy and the point 9 will be accurately determined. 

Again, from 1 measure off the departure of 9 at right angles 
to the assumed meridian, from the end of which, and at right 
angles to it, measure off the latitude ; the end of this latter line 
will be the required point 9. 

The surveyor should be satisfied with nothing less than per- 
fect coincidence of the points so determined. Fig. 186 shows 
sectional maps of work performed in this manner. 

399. To produce a line, previously marked out on the sur- 
face between two shafts, in the same relative direction below 
ground, so as to form a heading or tunnel from one shaft to 
another, proceed in the following manner : 

Set up the transit at the bottom and in the centre of the 
shaft, as near as can be estimated by the eye, as at a, Eig. 187, 
and, after the instrument has 

been accurately leveled and /^'^ ^^X^ 

the zero of the vernier made / 

to coincide with the zero of / 

the horizontal limb, sight -^ ^tz 
up the shaft and make the ^V 

cross- wires of the telescope \ 

bisect a mark J, in the line 
AB at the surface ; revolve 
the telescope vertically a little until the vertical wire strikes the 
opposite side of the shaft at c. Measure the angle cde, or its equal 



Fig. 187. 



cae 



-^ , and the radius of the shaft ae. The exact deviation of the 

centre of the transit from the vertical plane of the line AB at 
the surface, may then be found by calculation thus: Let the 



360 ELEMENTS OF SUEVEYING. [BOOK X. 

angular deviaJon, or angle cae, be 3° 10', and the radius of the 
shaft 60 inches, then 

nnp 

sin ^ X «e = .27639 x 60 = 16.5, 

the deviation required in inches. Remove the transit from a 
along the line ss' , and toward .s', a distance of 16.5 inches ; it 
will then occupy its true position, or be in the same vertical 
plane as the line AB at the surface. 

The transit may now be adjusted and leveled, and the tele- 
scope pointed up the shaft; when, if the preceding operation has 
been properly performed, the vertical wire will exactly pass 
through the marks previously fixed at e and b (H. D. Hoskold). 
Eevolving the telescope down, its line of sight will give the true 
direction below ground of the line AB at surface. 

Method of Plotting the Traverse on Paper. 

400. To plot the traverse on paper, we first plot the plan 
by the usual method of plotting compass-work, using the bear- 
ings and the reduced lengths of the courses. This gives the 
general direction of the horizontal projection of the traverse 
run ; and from the measurements for cross-section, the breadths 
of the tunnel on each side may be plotted, and thence a 
complete plan of the mine may be constructed. We next plot 
the profile of the traverse, using, as in railroad plotting, two 
scales, one for horizontal distances, and the other and larger 
one for vertical distances. The relation between the two scales 
will depend upon the circumstances of the case. Sometimes, 
both may be equal. The profile represents the undulation of 
the traverse, without reference to its horizontal deviations. Let 
us conceive vertical planes to be passed through all the courses. 
These will intersect each other in vertical lines. Take the one 
through the first course, as the one on which the profile is to 



SEC. IV.] 



PRACTICAL APPLICATIONS. 



361 



be delineated. Then, beginning with the plane through the 
last course, conceive the other planes to be revolved, in order, 
each about its intersection with the preceding one, to coincide 
with it, and so on till all are brought into coincidence with 
the fixed one. The lines of the traverse will then be situated 
in one plane, and a plot of them, in this position, will be the 
profile required. The distances from the traverse to the floor 
and roof of the tunnel, at different points, enable us to com- 
plete the profile. 



SECTION IV. 

PRACTICAL APPLICATIONS. 

401. To determine the position and depth of a mine shaft, 
CB, Fig. 188, which is to connect with a tunnel AB, 




Fig. 188. 

If the course of the tunnel is not a straight line it must be 
traversed. Assuming it to be a straight line, the bearing is 
obtained with the transit. The length of the tunnel is then 
accurately measured with a steel tape or a rod. The surveyor 
DOW stakes out the course on the surface, then beginning at the 



362 



ELEMENTS OF SURVEYING. 



[book X. 



mouth of the tunnel the first peg is driven on a level with the 
door, and the measurement of tlie tunnel, say 1000 feet, is laid 
off on the course established on the surface. If the slope rod is 
used, the sum of the vertical measurements or readings will be 
the vertical distance of (7 above A. Should the tunnel at B be 
higher or lower than at A, the difference subtracted from or 
added to the sum of the vertical readings, will be the required 
depth of the shaft. If the slope-rod is not used, a line of levels 
must be run to find the difference of level between A and C. 

402. To find the distance necessary to connect the main 
shaft GFsmd the tunnel FF (Fig. 189). 




Fig. 189. 

Determine the depth of the shaft BF, by suspending a chain 
from the surface or by a steel tape with a weight attached. 
Measure the length of the tunnel FH^ already completed, and, 
with the slope rod, or otherwise, obtain the horizontal distance 
DI, and the difference of level between the points I) and G. 
This difference of level added to the depth of the shaft DF ■= the 
total depth of the shaft GF. Find the depth of the shaft GJ 
already completed. Then GF— GJ = JF, and FF—FH = HF, 
the required distances. In this case the tunnel is assumed to be 



SEC. IV.] 



PRACTICAL APPLICATIONS. 



363 



straight and the position of the shaft determined as in the 
preceding problem. 

403. To find the depth of a mine shaft AB, Fig. 190, the 
distance from the outcrop at C and the dip of the vein CB 
being given. 

Determine the difference 
of level AD = 100 ft. 

The horizontal distance 
CD = 600 ft. 

Angle DCB = 60°. 

Then DB = CD tan 60° 
= 1039.2 ft. 

And AB = AD + DB 
= 100 4- 1039.2 = 1139.2. 

To find the distance 
from the outcrop to the 
point of intersection, or the 
distance CB, we have (Da- 
vies' Leg., Trig., Art. 37), 

COS 60 ^ 

BY CONSTRUCTION. 

Draw to any suitable scale the distance CD = 600 feet ; from 
C draw the line CD, making an angle of 60° with the line CD ; 
from the point D draw a line perpendicular to CD, until it inter- 
sects CB at B; produce it upward to A, until distance from D 
to ^ = difference of elevation between C and A. Measure the 
line ^^ on the scale, which gives the required depth. Measure 
also CB on the scale, which is the distance from the outcrop to 
the point of intersection. 

404. Given the depth of the shaft DB = 588.5 feet. Fig. 
191, and the horizontal distance FD from the outcrop to the top 




364 



ELEME]S"TS OF SURVETIJs^G. 



[book X. 



of shaft = 500 feet, and the angle GFD = 60°, to find the 

length of a cross-cut UG 

from*'' the bottom of the 

shaft to intersect the vein ; 

also the distance from the 

outcrop to the point of 

intersection. 

Find the horizontal 
distance from the outcrop 
F, at which a shaft of 
588.5 feet will intersect 
the vein, or the distance 
FH. This is readily found; 
for assuming it to be the 
base of a right-angled tri- 
angle, we have (Davies' Leg., Trig., Art. 37), 




Fig. 191. 



FII = 



588.5 
tan 60' 



339.78. 



If a depression or elevation from the outcrop occurs at the point, 
it must be subtracted from or added to the depth of the shaft ; 
for example, if there is a depression of 50 ft. it will only be neces- 
sary to sink 538.5 ft. to the point of intersection. The horizontal 
distance so found, subtracted from the real distance of the shaft 
from the outcrop = length of cross-cut = 160.22, and 

FG = VDE^ + {FD - GEf = 679.5. 



BY CONSTRUCTION. 



Draw the line FD = 500 feet to any convenient scale. From 
F draw the line FG, making the angle DFG = 60°. Draw 
DF = 588.5 ft., and from F draw FG parallel to FD. The dis- 
tances can now be taken from the scale. 



SEC. IV.] 



PRACTICAL APPLICATION'S. 



365 



405. Given the angle LJK = 60°. The distance JL=1!00 ft., 
and the difference of level between J and ff = 100 ft. Required 
the depth of a shaft from 5" that will intersect a cross-cut of 350 ft. 
from the point K, Fig. 192. 




Fig. 192. 

From the point K draw to JL a line parallel to III ; it "will 
intersect JL 700—350 = 350 ft. from the point /. This line is 
the perpendicular of a right-angled triangle, and we have 
it = 350 tan 60°. Add to this the difference of elevation, 
which is the required depth. 



BY CONSTRUCTION. 

Draw the line JX=700 ft., and from / draw JE, making the 
angle LJIC = 60°. From K set off the distance KI, parallel to 
JL, and make it 350 ft. From / raise the perpendicular IL and 
extend it to JI, niaking the distance LH = 100 ft. of the scale. 
The distance IH is now taken from the scale. 

Problems of this kind occur only where there are inclined 
shafts. 



366 



ELEMENTS OF SURVEYING. 



[book X. 



406. Given the distance OL = 400 ft., and the angle LON= 
60° ; to find the depth of a shaft at the point of intersection 
with the vein, and to find the length 
of a cross-cut that will again inter- 
sect the vein when the shaft is 
continued 250 ft. below the point of 
its intersection, Fig. 193. 

Draw the horizontal line OL 
= 400 ft. Also ON, making the 
angle LOW = 60°. Then 400 
tan 60° = depth of shaft at the 
point of intersection. Construct 
the shaft and extend it to the point 
Jf, 250 ft. below the point of inter- 
section, and draw the cross-cut 
MW. Then by similar triangles, 




M 



A" 



Fig. 193. 



OL : LP :: MN : PM\ 
OL X PM 



MN = 



LP 



BY CONSTRUCTION. 

Draw OL = 400 ft.; also ON, making the angle LO]Sr=60° ; 
and LM to the point M, 250 ft. below the point of intersection P. 
From the scale take off LP and MNthe required distances. 

In the preceding problems the angle of 60° was employed for 
convenience of construction, but the principles would l)e true for 
any other angle. 

407. Figures 194, 195, 196, represent plan, longitudinal 
and transverse sections of a developed mine. 

The development consists of a vertical shaft, an inclined 
shaft following the dip of the vein, and 6 one-hundred-foot 
levels. 



SEC. IV.] 



PRACTICAL APPLICATIONS. 



367 




368 



ELEMENTS OF SURYETING. 



[book X. 



In opening mines it is considered good practice to follow the 
vein for a considerable distance in depth, to be fully satisfied of 
its continuity, before sinking a vertical shaft for deep working. 

Should the vein be vertical, however, the prospecting shaft 
may be made the working shaft. The depth to which a shaft is 
sunk on the dip of the vein will depend upon the Engineer in 




Fig 19c. 

charge and upon the characteristics of the vein. A careful 
engineer will not incur the expense of sinking a deep working shaft 
to intersect an inclined vein, until he has followed the vein for 
such a distance that the possibility of its terminating, or " pinch- 
ing out," is quite remote. Should the vein be irregular and not 
possess well-defined fissure characteristics, the greater the neces- 
sity for care in this respect and, in cases of great uncertainty, it 
would even be advisable to have the working shaft follow the dip 



SEC. rv.] 



PEACTICAL APPLICATIONS. 




Fig. 196. 



370 ELEMENTS OF SUEVEYIJ^G. [BOOK X. 

of the vein. If, on the other hand, the vein is found to be strong 
and regular, possessing well-defined fissure characteristics to a 
depth of 150 to 250 feet, the working vertical shaft might be 
sunk with comparative safety. No rule can be laid down, how- 
ever, and the Engineer must always exercise his own judgment. 

In the example here given, we have assumed the inclined shaft 
to be sunk to the same depth as the vertical, for if originally 
sunk to the first cross-cut, 440 feet (Fig. 196), it would be car- 
ried down to the point of intersection and used for ventilation. 

The method of surveying and plotting such a mine is simply 
an application of the principles already explained. 

408. The calculation of ore-reserves does not come strictly 
within the province of the surveyor, yet after completing the 
survey and plot, he is frequently required to make the calculation. 
We will therefore consider methods of making it. 

In practice the methods employed are various. No general 
rule can be given, as each expert has a system of his own, and dif- 
ferent engineers will not agree, within wide limits, as to the 
quantity of ore-reserves in the same mine. One may assume as 
the measure of the ore in sight a rectangular block limited by 
the outcrop, the depth of the shaft or shafts, and the extreme 
points of the levels, diminished by the amount extracted. 
Others, but one-half or one-third of this quantity. The former 
would be considered an excessive estimate in all cases. The lat- 
ter too low when the vein possesses great strength and regularity, 
though even this estimate may be too high, when the conditions 
are the reverse. The surveyor must exercise his own judgment, 
exercising caution, however, as the calculation is an important 
one. 

Assume the development shown in Figs. 194, 195, 196, and 
let it be required to calculate the ore-reserves when the bounding 
lines are assumed at the extreme ends of the level drifts, or the 



SEC. IV.] PRACTICAL APPLICATION'S. 371 

rectangular block ABCD, Fig. 195, and when the average cross- 
section of the vein is 6 feet, and a cubic foot of the vein matter 
in place weighs 150 lbs. 

Ore stopes, or steps made in mine- workings by and for 
the extraction of ore, are generally very irregular, the repre- 
sentation here being an ideal one. Suppose the stope-faces to be 
11 feet apart and 8 feet high, and that the inclined shaft has 
extracted 10 x 6 ft. of vein matter, and the levels 7 x 6 f t. 

We see that the inclined shaft has exposed the vein for 

440 -f 115 + 115 = 670 ft.; deducting say 15 ft. for inequality of 

surface, we should have a rectangular block 655 x (400 + 350) 

X 6 in width = 2947500 cubic feet of ore: to be deducted from 

this, we have 

The inclined shaft 655 x 10 x 6 = 39300 cubic feet. 

1st level (150 + 200) x 7 x 6 

2d '' 260 X 7 X 6 

3d '' 520 X 7 X 6 

4th '' 240 X 7 X 6 

5th '' 345 X 7 X 6 

6th " 500 X 7 X 6 

Stoped out on 1st level east, roughly estimated, 3400 



iC li 


west 


a 


2d " 


ti 


i( 


3d " 


east 


It 


6th " 


west 


<( 



= 14700 




= 10920 




= 21840 




= 10080 




= 14490 




= 21000 




nated, 3400 




'• 6500 




7000 




« 20000 




12000 





Total, 182000 cu. ft. (say). 

Deducting this from 2947500, we have, 2765500 " '' 

Dividing by 13^, the number of cu. ft. required for a ton, and we 
have 204852 tons of ore in sight. 

Another method of calculation is as follows: The longest 



372 ELEMEI^TS OP SURVEYING. [BOOK X. 

drift east is 400 ft. and the shortest 100 ft. Assume the bounding 
line in this direction to be at a distance east of the shaft. 

The longest drift west is 350 ft. and the shortest 100 ft.; take the 
bounding line in this direction at a distance 

100 + ^^» - ^»» ^ m ft., 

or the rectangular block alcF, Fig. 195. Calculate the ore- 
reserves, when the other data are the same as before. 

This latter method is recommended by competent engineers 
as the fairest and most reliable for all parties concerned. 

409. Fig. 197 shows a longitudinal section, and Fig. 198 a 
transverse section of a deposit mine with mill connections, the 
mill to be erected at the point A. From a consideration of the 
diagram, it is evident that the most convenient method for the 
transportation of the ore from the mine to the mill would be by 
a tunnel driven into the mountain, at the end of which is a bin, 
made in the solid rock and inclined to the tunnel at any convenient 
angle at which ore will slide into cars ; the cars to be run into the 
tunnel on a track and directly under iron doors which are worked 
by rack and pinion. The bin is to connect with the ore-chamber 
by a chute inclined at an angle of 45°, as shown in the diagram. 

The lower or mill tunnel should have a slope of 2 inches in 
10 leet, so that the loaded cars would descend by the force of 
gravity, the last car in a train having a brake with which to 
regulate the speed. The chute should be 12 to 15 feet from the 
edge of the tunnel, to admit of constructing the inclined bin for 
the discharge of the ore into the cars. The point in the ore- 
chamber, at which it is desired to sink the chute, and the mouth 
of the lower tunnel being selected, drive a peg to the centre point 



SEC. IV.] 



PRACTICAL APPLICATIONS. 



373 




374 ELEME2>'TS OF SUEYEYIi^G. [BOOK X. 

of the proposed tunnel floor and drive a nail in the peg, and 
rei^eat the operation at the point where the chute is to be sunk. 
Xow make a careful traverse between these points ; the direction 
of a line which will run from the mouth of the tunnel directly 
under the point selected for the chute can now be found, as 
explained in the section on traversing, and the course that will 
carry the tunnel 12 or 15 feet from the bottom of the chute may 
be determined. In driving the tunnel, holes should be drilled in 
the roof and wooden spuds driven in on which to hang plumb-bobs, 
the surveyor using great care to have the plumb-bobs suspended 
in the proper course as a guide to the miners. In starting the 
chute, a large ^rooden triangle should be made, one of the angles 
of which is the same as the angle of the chute, to be used 
by the miners as a guide, until sufficient depth is attained to 
hang, in the proper line, plumb-bobs, the points of which are on 
the required angle. 



APPENDIX A. 



THE SOLAR COMPASS. 

( With some omissions, from Messrs. W. and L. E. Ourley's Manual of Engi- 
neering and Surveying Instruments, ^I^th Edition, 18S3.) 

This instrument, so ingeniously contrived for readily determining a true 
meridian or north and south line, was invented by William A, Burt, of Mich- 
igan, and patented by him in 1836. 

It has since come into general use in the surveys of U. S. public lands, 
the principal lines of which are required to be run with reference to the true 
meridian. 

The arrangement of its sockets and plates is similar to that of the Survey- 
ors' Transit, except that the sight vanes are attached to the under plate or 
limb, and this revolves around the upper or vernier plate on which tlie solar 
apparatus is placed. 

The limb is divided to half degrees, is figured in two rows, and reads by 
the two opposite verniers to single minutes. 

The Solar Apparatus.— The Solar Apparatus is seen. Fig. 1, in 
the place of the needle, and in fact operates as its substitute in the field. 

It consists mainly of three arcs of circles, by which can be set off the lati- 
tude of a place, the declination of the sun, and the hour of the day. 

These arcs, designated in the cut by the letters a, b, and c, are therefore 
termed the latitude, the declination, and the hour arcs respectively. 

The Latitude Arc, a, has its centre of motion in two pivots, one of 
which is seen at d, the other is concealed in the cut. 

It is moved either up or down within a hollow arc, seen in the cut, by a 
tangent screw at/, and is securely fastened in any position by a clamp-screw 

The Latitude arc is graduated to quarter degrees, and reads by its vernier, 
^5, to single minutes ; it has a range of about thirty-five degrees, so as to be 
adjustable to the latitude of any place in the United States. 

The Declination Arc, &, is also graduated to quarter degrees, and 
has a range of about twenty-eight degrees. 

Its vernier, v, reading to single minutes, is fixed to a movable arm, h, hav- 
ing its centre of motion at the end of the declination arc at g ; the arm is 
moved over the surface of the declination arc, and its vernier set to any read- 
ing by turning the head of the tangent screw k. It is also securely clamped 
in any position by a screw, concealed in the engraving. 

Solar Lenses and Lines. — At each end of the arm, h, is a rectangu- 
lar block of brass, in which is set a small convex lens, having its focus on the 



ELEMENTS OF SURVEYI2q"G. 



[app. a. 



surface of a little silver plate A, Fig. 2, fastened by screws to the inside of 
the opposite block. 




On the surface of the plate are marked two sets of 
lines intersecting each other at right angles ; of these h b 



are termed the hour lines, and c c the equatorial lines,, as "/> rrj 
having reference respectively to the hour of the day and ^ ' ■ 

the position of the sun in relation to the equator. Fig. 2. 

In Fig. 1 the equatorial lines are those on the lower block, parallel to the 
surface of the hour arc c ; the hour lines are of course those at right angles 
to the first. 

Equatorial Sights. — On the top of each of the rectangular blocks 
is seen a little sighting-piece, termed the equatorial sight, fastened to the 
block by a small milled head-screw, so as to be detached at pleasure. 



A pp. A.] THE SOLAR COMPASS. 3 

They are used, as will be explained hereafter, in adjusting the different 
parts of the solar apparatus. 

The Hour Arc, c, is supported by the two pivots of the latitude arc 
already spoken of, and is also connected with that arc by a curved arm, as 
shown in the figure. 

The hour arc has a range of about 120°, is divided to half degrees, and 
figured in two series ; designating both the hours and the degrees, the middle 
division being marked 12 and 90 on either side of the graduated lines. 

The Polar Axis. — Through the centre of the hour arc passes a hollow 
socket, p, containing the spindle of the declination arc, by means of which 
this arc can be moved from side to side over the surface of the hour arc, or 
turned completely round, as may be required. 

The hour arc is read by the lower edge of the graduated side of the decli- 
nation arc. 

The axis of the declination arc, or indeed the whole socket p, is appropri- 
ately termed the polar axis. 

The Adjuster. — Besides the parts shown in the cut, there is also an 
arm used in the adjustment of the instrument as described hereafter, but laid 
aside in the box when that is effected. 

The parts just described constitute properly the solar apparatus. 

Besides these, however, are seen the needle-box, n, with its arc and tan- 
gent screw, t, and the spirit levels, for bringing the whole instrument to a 
horizontal position. 

The Needle-Box n, has an arc of about 36" in extent, divided to half 
degrees, and figured from the centre or zero mark on either side. 

The needle is raised or lowered by a lever shown in the cut. 

The needle-box is attached by a projecting arm to a tangent- screw, t, by 
which it is moved about its centre, and its needle set to any variation. 

This variation is also read off by the vernier on the end of the projecting 
arm, reading to three minutes a graduated arc, attached to the plate of the 
compass. 

The Levels seen with the solar apparatus have ground glass vials, and 
are adjustable at their ends like those of other instruments. 

The edge of the circular plate on which the solar work is placed, is divided 
and figured at intervals of ten degrees, and numbered, as shown, from to 90 
on each side of the line of sight. 

These graduations are used in connection with a little brass pin, seen in 
the centre of the plate, to obtain approximate bearings of lines, which are 
not important enough to require a close observation. 

Lines of Refraction —The inside faces of the sights are also gradu- 
ated and figured, to indicate the amount of refraction to be allowed when the 
sun is near the horizon. These are not shown in the cut. 

Principles of the Solar Compass. — The interval between two 

equatorial lines c c, in Fig. 2, as well as between the hour lines 6 h, is just 
sufficient to include the circular image of the sun as formed by the solar lens 
on the opposite end of the revolving arm h, Fig. 1. 

When, therefore, the instrument is made perfectly horizontal, the equa- 



4 ELEMENTS OF SURVEYING. [aPP. A. 

torial lines and the opposite lenses being accurately adjusted to each other 
by a previous operation, and the sun's image brought within the equatorial 
lines, his position in the heavens, with reference to the horizon, will be de- 
fined with precision. 

Suppose the observation to be made at the time of one of the equinoxes ; 
the arm h, set at zero on the declination arc h, and the polar axis p placed 
exactly parallel to the axis of the earth. 

Then the motion of the arm h, if revolved on the spindle of the declination 
arc around the hour circle c, will exactly correspond with the motion of the 
sun in the heavens, on the given day and at the place of observation ; so that 
if the sun's image was brought between the lines c c, in the morning, it 
would continue in the same position, passing neither above nor below the 
lines, as the arm was made to revolve in imitation of the motion of the sun 
about the earth. 

In the morning, as the sun rises from the horizon, the arm h will be in a 
position nearly at right angles to that shown in the cut, the lens being turned 
towards the sun, and the silver plate on which his image is thrown directly 
opposite. 

As the sun ascends, the arm must be moved around, until, when he has 
reached the meridian, the graduated side of the declination arc will indicate 
12 on the hour circle, and the arm h, the declination arc h, and the latitude 
arc a, will be in the same plane. 

As the sun declines from the meridian the arm 7i must be moved in the 
same direction, until at sunset its position will be the exact reverse of that it 
occupied in the morning. 

Allowance for Declination. — Let us now suppose the observation 
made when the sun has passed the equinoctial point, and when his position 
is affected by declination. 

By referring to the Almanac, and setting off on the arc his declination for 
the given day and hour, we are still able to determine his position with the 
same certainty as if he remained on the equator. 

When the sun's declination is south, that is, from the 22d of September 
to the 20th of March in each year, the arc 6 is turned toward the plates of the 
compass, as shown in the engraving, and the solar lens, o, with the silver 
plate opposite, are made use of in the surveys. 

The remainder of the year, the arc is turned from the plates, and the 
other lens and plate employed. 

When the Solar Compass is accurately adjusted, and its plates made per- 
fectly horizontal, th- latitude of the place, and the declination of the sun for 
the given day and hour, being also set off on the respective arcs, the image 
of the sun cannot he brought betioeen the equatorial lines until the polar axis 
is placed in the iMne of thz meridian of the place, or in a position parallel to 
Ihi axis of the earth. The slightest deviation from this position will cause 
the image to pass above or below the lines, and thus discover the error. 

We thus, from the position of the sun in the solar system, obtain a certain 
direction absolutely unchangeable, from which to run our lines, and measure 
the horizontal angles required. 

This simple principle is not only the basis of the construction of the Solar 
Compass, but the sole cause of its superiority to the ordinary or magnetic in- 
strument. For in a needle instrument the accuracy of the horizontal angles 



APP. A.] THE SOLAR COMPASS. 5 

indicated, and tlieref ore of all tlie observations made, depends upon •' the deli- 
cacy of the needle, and the constancy with which it assumes a certain direc- 
tion, termed the magnetic meridian." 

The principal causes of error in the needle, briefly stated, are the dulling 
of the pivot, the loss of polarity in the needle, the influence of local attraction, 
and the effect of the sun's rays, producing the diurnal variation. 

From all these imperfections the solar instrument is free. 

The sights and the graduated limb being adjusted to the solar apparatus, 
and the latitude of the place and the declination of the sun also set off" upon 
the respective arcs, we are able not only to run the true meridian, or a due 
east and west course, but also to set off the horizontal angles with minuteness 
and accuracy from a direction which never changes, and is unaffected by at- 
traction of any kind. 

To Adjust the Solar Compass.— The adjustments of this in- 
strument, with which the surveyor will have to do, are few in number, and 
will now be given in order. 

1st. To Adjust the Levels. — Proceed precisely as directed in the 

account of the other instruments described, by bringing the bubbles into the 
centre of the tabes by the leveling screws of the tripod, and then reversing 
the instrument upon its spindle, and raising or lowering the ends of the tubes 
until the bubbles will remain in the centre during a complete revolution of 
the instrument. 

2d. To Adjust the Equatorial Lines and Solar Lenses — 

First detach the arm 7i from the declination arc by withdrawing the screws 
shown in the cut from the ends of the posts of the tangent-screw k, and also 
the clamp-screw, and the conical pivot with its small screws by which the arm 
and declination arc are connected. 

The arm h being thus removed, attach the adjuster in its place by replac- 
ing the conical pivot and screws, and insert the clamp-screw so as to clamp 
the adjuster at any point on the declination arc. 

No\y level the instrument, place the arm 7i on the adjuster, with the same 
side resting against the surface of the declination arc as before it was detached. 
Turn the instrument on its spindle so as to bring the solar lens to be adjusted 
in the direction of the sun, and raise or lower the adjuster on the declination 
arc, until it can be clamped in such a position as to bring the sun's image as 
near as may be between the equatorial lines on the opposite silver plate, and 
bring the image precisely into position by the tangent of the latitude arc or 
the leveling-screws of the tripod. Then carefully turn the arm half way 
over, until it rests upon the adjuster by the opposite faces of the rectangular 
blocks, and again observe the position of the sun's image. 

If it remains between the lines as before, the lens and plate are in adjust- 
ment ; if not, loosen the three screws which confine the plate to the block, 
and move the plate under their heads, until one-half the error in the position 
of the sun's image is removed. 

Again bring the image between the lines, and repeat the operation until it 
will remain in th.e same situation, in both positions of the arm, when the ad- 
justment will be completed. 

To adjust the other lens and plate, reverse the arm end for end on the ad 



EBi 



6 ELEMENTS OF SURVEYING. [APP. A. 

juster, and proceed precisely as in the former case, until the same result is 
attained. 

In tightening the screws over the silver plate, care must be taken not to 
move the plate. 

This adjustment now being complete, the adjuster should be removed, 
and the arm h, with its attachments, replaced as before. 

3d. To Adjust the Vernier of the Declination Arc— Hav- 
ing leveled the instrument, and turned its lens in the direction of the sun, 
clamp to the spindle, and set the vernier 'o, of the declination arc, at zero, by 
means of the tangent-screw at k, and clamp to the arc. 

See that the spindle moves easily and yet truly in the socket, or polar 
axis, and raise or lower the latitude arc by turning the tangent-screw/, until 
the sun's image is brought between the equatorial lines on one of the plates. 
Clamp the latitude arc by the screw, and bring the image precisely into posi- 
tion by the level ing-screws of the tripod or socket, and without disturbing 
the instrument, carefully revolve the arm 7i, until the opposite lens and plate 
are brought in the direction of the sun, and note if the sun's image comes be- 
tween the lines as before. 

If it does, there is no index error of the declination arc ; if not, with the 
tangent-screw h, move the arm until the sun's image passes over half the 
error ; again bring the image between the lines, and repeat the operation as 
before, until the image will occupy the same position on both the plates. 

We shall now find, however, that the zero marks on the arc and the ver- 
nier do not correspond, and to remedy this error, the little flat-head screws 
above the vernier must be loosened until it can be moved so as to urake the 
zeros coincide, when the operation will be completed. 

4th. To Adjust the Solar Apparatus to the Compass 

Sights. — First, level the instrument, and with the clamp and tangent- 
screws set the main plate at 90° by the verniers and horizontal limb. Then 
remove the clamp-screw, and raise the latitude arc until the polar axis is by 
estimation very nearly horizontal, and if necessary, tighten the screws on the 
pivots of the arc, so as to retain it in this position. 

Fix the vernier of the declination arc at zero, and direct the equatorial 
sights to some distant and well-marked object, and observe the same through 
the compass sights. If the same object is seen through both, and the ver- 
niers read to 90° on the limb, the adjustment is complete ; if not, the correc- 
tion must be made by moving the sights or changing the position of the. 
verniers. 

It should be remarked that as the solar work is attached permanently to 
the sockets, and this adiustment is made by the maker, it will need no fur- 
ther attention at the hands of the surveyor except in case of serious accidents. 

The other adjustments are of course also made in the. process of finishing 
the instrument, and are liable to very little derangement in the ordinary use 
of the Solar Compass. 

To Use the Solar Compass.— Before this instrument can be used 
at any given place, it is necessary to set off upon its arcs both the declination 
of the sun as affected by its refraction for the given day and hour, and the 
latitude of the place where the observation is made. 



APP. A.] THE SOLAR COMPASS. 7 

To Set Off the Declination. — The declination of the sun, given 
in the ephemeris of the Nautical Ahnanac from year to year, is calculated for 
apparent noon at Greenwich, England. 

To determine it for any other hour at a place in the U. S., reference must 
be had, not only to the difference of time arising from the longitude, but also 
to the change of declination from day to day. 

The longitude of the place, and therefore its difference in time, if not 
given directly in the tables of the Almanac, can be ascertained very nearly 
by reference to that of other places given, which are situated on, or very 
nearly on, the same meridian. 

It is the practice of surveyors in the States east of the Mississippi to allow 
a difference of six hours for the difference in longitude, calling the declination 
given in the Almanac for 13 M., that of 6 A. M., at the place of observation. 

Beyond the meridian of Santa Fe, the allowance would be about seven 
hours, and in California, Oregon, and Washington Territory about eigJit hours. 

Having thus the difference of time, we very readily obtain the declination 
for a certain hour in the morning, which would be earlier or later as the. lon- 
gitude was greater or less, and the same as that of apparent noon at Green- 
wich on the given day. Thus, suppose the observation made at a place, say, 
five hours later than Greenwich, then the declination given in the Almanac 
for the given day at noon, affected by the refraction, would be the declination 
at the place of observation for 7 o'clock A. M. ; this gives us the starting- 
point. 

To obtain the declination for the other hours of the day, take from tlie 
Almanac the declination for apparent noon of the given day, and, as the dec- 
lination is increasing or decreasing, add to or subtract from the declination of 
the first hour the difference for one hour as given in the ephemeris, which 
will give, when affected by the refraction, the declination for the succeeding 
hour ; and proceed thus in making a table of the declination for every hour 
of the day. 

To Set Off the Latitllde. — Find the declination of the sun for the 
given day at noon, at the place of observation as just described, and with the 
tangent-screw set it off upon the declination arc, and clamp the arm firmly to 
the arc. 

Observe in the Almanac the equation of time for the given day, in order to 
know about the time the sun will reach the meridian. 

Then, about fifteen or twenty minutes before this time, set up the instru- 
ment, level it carefully, fix the divided surface of the declination arc at 12 on 
the hour circle, and turn the instrument upon its spindle until the solar lens 
is brought into the direction of the sun. 

Loosen the clamp-screw of the latitude arc, and with the tangent-screw 
raise or lower this arc until the image of the sun is brought precisely between 
the equatorial lines, and turn the instrument from time to time so as to keep 
the image also between the hour lines on the plate. 

As the sun ascends, its image will move below the lines, and the arc must 
be moved to follow it. Continue thus, keeping it between the two sets of 
lines until its image begins to pass above the equatorial lines, which is also 
the moment of its passing the meridian. 

Now read off the vernier of the arc, and we have the latitude of the place. 



tel 



8 ELEMENTS OP SUEVEYIKG. [aPP. A. 

which is always to be set off on the arc when the compass i& used at the 
given place. 

It is the practice of surveyors using the Solar Compass to set off, in the 
manner just described, the latitude of the point where the survey begins, and 
to repeat the observation and correction of the latitude arc every day when 
the weather is favorable, there being also nearly an hour at mid-day when 
the sun is so near the meridian as not to give the direction of lines with the 
certainty required. 

To Run Lines with the Solar Compass.— Having set off in 

the muuuer just given the latitude and declination upon their respective 
arcs, the instrument being also in adjustment, the surveyor is ready to run 
lines by the sun. 

To do this, the instrument is set over the station and carefully leveled, 
the plates clamped at zero on the horizontal limb, and the sights directed 
north and south, the direction being given, when unknown, approximately by 
the needle. 

The solar lens is then turned to the sun, and with one hand on the in- 
strument, and the other on the revolving arm, both are moved from side to 
side, until the sun's image is made to appear on the silver plate ; when by 
carefully continuing the operation, it may be brought precisely between the 
equatorial lines. 

Allowance being made for refraction, the line of sights will indicate the 
true meridian ; the observation may now be made, and the flag-man put in 
position. 

When a due east and west line is to be run, the verniers of the horizontal 
limb are set at 90°, and the sun's image kept between the lines as before. 

The Solar Compass being so constructed that when the sun's image is in 
position the limb must be clamped at in order to run a true meridian line, 
it will be evident that the bearing of any line from the meridian may be read 
by the verniers of the limb, precisely as in the ordinary magnetic compass 
the bearing of lines are read from the ends of the needle. 

Use of the Needle. — in running lines, the magnetic needle is always 
kept with the sun ; that is, the point of the needle is made to indicate on 
the arc of the compass-box, by turning the tangent-screw connected with its 
arm on the opposite side of the plate. By this means the lines can be run by 
the needle alone in case of the temporary disappearance of the sun ; but, of 
course, in such cases the surveyor must be sure that no local attraction is 
exerted. 

The variation of the needle, which is noted at every station, is read off in 
degrees and minutes on the arc, by the edge of which the vernier of the 
needle-box moves. 

Allowance for the Earth's Curvature.— When long lines are 

run by the Solar Compass, either by the true meridian, or due east and west, 
allowance must be made for the curvature of the earth. 

Thus, in running north or south, the latitude changes about one minute 
for every distance of 92 chains 30 links, and the side of a township requires 
a change on the latitude arc of 5' 12", the township, of course, being six miles 
square. 



APP. A.] THE SOLAR COMPASS. 9 

This allowance is of constant una where tho surveyor fails to got an obser- 
vation on the sun at noon, and is a very close approximation to the truth. 

In running due east and vveHt, as in tracing the standard parallels of lati- 
tude, the sights are set at 90 on the limb, and the line is run at right angles to 
the meridian. 

If no allowance were raxule for the earth's curvature, these lines would, if 
8ulHcif;ntly produrx-d, reach tlie e'piator, to which thoy are constantly tending. 

Of Ofmrfid, in running short lines either east or wefrt, the variation from 
the parallel would be so small as to be of no jjractical imwrtance ; but when 
long siglits are taken, the c^^rroction should be made by taking fore and back 
sights at every station, noticing the error on the ba^,*k sight, and sftting off 
onohalf of it on the fore sight on the side towards the pole. 

Time of Day by the Sun.— The time of day is best ascertained by 
the Solar Compass when the sun is on the meridian, as at the time of making 
the observation for latitude. 

The time thus given is that of apparent nrx)n, and can be reduced to mean 
time by merely applying the equation of time as directed in the Almanac, and 
adding or subtracting as the sun is slow or fast. 

The time, of course, can also be taken before or after noon, by bringing 
the sun's image between the hour lines, and noticing the position of the 
divided edge of the revolving arm, with referenc<i to tlie graduations of the 
hour circle, allowing four minutes of time for each degree of the arc, and thus 
obtaining apparent time, which must be correctf>d by the equation of time as 
just decribed. 

Caution as to the False Image.— in using the compass upon the 

iun, if the revolving arm be turned a little one side of its proper position, a 
false or reflected image of the sun will apyjear on the silver j)late in nearly 
the same place as that fxxupied by the true one. It is caused by the reflec- 
tion of the true image frrmi the surface of the arm, and is a fruitful source 
of error to the inexperienf;ed surveyor. It can, however, be readily distin- 
guished from the real image by being much less bright, and not so clearly 
defined. 

Approximate Bearings. — When the bearin/Brs of lines, such as the 
course of a stream, or the Fxjundaries of a forest, are not desired with the cer. 
tainty given by the verniers and horizontal limb, a rough approximation of 
the angle they make with the true meridian is obtained by the divisions on 
the outside of the circular plate, 

In this ojxjration, a pencil, or thin straight edge of any sort, is held per- 
pendicularly against the circular edge of the plate, and moved around until it 
is in range with the eye, the brass centre-pin, and the object observed. 

The bearing of the line is then read off' at the point where the pencil is 
placed. 

Time for Using the Solar Compass.— The Solar Compass, like 

the ordinary instrument, can be usf^d ut all seasons of the year, the most favor- 
able time l>eing, of course, in the summer, when the declination is nortli, and 
the days are long, and more generally fair. 

It is best not to take the sun at morning and evening, when it is within 



lO 



:e:lements of surveyin'G. 



[app. a. 



half an hour of the horizon, nor at noon, for about the same interval, before 
and after it passes the meridian. 

Allowance for Refraction. — The proper allowance to be made for 
refraction in setting oflPthe declination of the sun upon the Solar Compass has 
long been a source of perplexity to the surveyor ; we have, accordingly, given 
the subject a good deal of attention, and here publish a table, by which the 
amount of refraction for any hour of any day of the year can be ascertained, 
and set off with a degree of accuracy which is all that can be desired. 

A TABLE OF MEAN REFRACTIONS IN DECLINATION. 

To apply on the declination arc of Solar Attachment of either Compasses 
or Transits. 

Computed by Edward W. Arms, C. E., for W. & L. E. Gurlet, Troy, 
N. Y. 



1^ 
o 


DECLINATIONS. 


For Latitude 30°. 


+ 20° 


+ 15° 


+ 10° 


+ 5° 


0° 


5° 


—10° 


—15° 


—20° 


Oh. 


10" 


15'' 


21" 


27" 


33'' 


40" 


48" 


57" 


1'08" 


2 


14 


19 


25 


31 


38 


46 


54 


T05 


118 


3 


20 


26 


32 


39 


47 


55 


1'06 


119 


136 


4 


32 


39 


46 


52 


1^06 


1'19 


135 


157 


2 29 


5 


I'OO 


I'lO 


r24 


r52 


2 07 


244 


3 46 


5 43 


13 06 


For Latitude 33° 30' . 


Oh. 


13" 


18" 


24" 


30" 


36" 


44" 


52" 


1'02" 


1'14" 


2 


17 


22 


28 


35 


42 


50 


I'OO 


111 


126 


3 


23 


29 


35 


43 


51 


I'Ol 


113 


128 


147 


4 


35 


43 


51 


roi 


1'13 


127 


146 


2 13 


2 54 


5 


1'03 


1'15 


1'31 


153 


2 20 


3 05 


4 25 


7 36 




For Latitude 35°. 


Oh. 


15" 


21" 


27" 


33" 


40" 


48" 


57" 


1'08" 


1'21" 


2 


20 


25 


32 


38 


46 


55 


105 


1 18 


135 


3 


26 


33 


39 


47 


56 


1'07 


121 


1 38 


2 00 


4 


39 


47 


56 


1'07 


1'20 


136 


159 


2 32 


3 25 


5 


1'07 


1'20 


1'38 


2 00 


2 34 

1 


3 29 


5 14 


10 16 


1 


For Latitude 37° 30'. 


Oh. 


18" 


24" 


30" 


36" 


44" 


52" 


1'02" i 


1'14" 


1'29" 


o 


22 


28 


35 


42 


50 


I'OO 


112 


126 


145 


3 


29 


36 


43 


52 


1'02 


114 


129 


149 


2 16 


4 


43 


51 


I'Ol 


1'13 


127 


149 


2 14 


2 54 


4 05 


5 


I'll 


1'26 


154 


2 10 


2 49 


3 55 


6 15 j 


14 58 








APP. A.] 



THE SOLAE COMPASS. 



11 









DECLINATIONS 


, 






< 






Fob Latitude 40°. 




o 


+ 20° 


+ 15° 


+ 10° 


+ 5° 


0° 

48" 


-5° 


— 10° 
1'08" 


-15° 


-20 


Oh, 


21" 


27" 


33" 


40'' 


57" 


1'21" 


1'33" 


2 


25 


32 


39 


46 


52 


1'06 


119 


135 


157 


8 


33 


40 


48 


57 


ro8 


121 


138 


2 02 


2 36 


4 


47 


55 


1'06 


119 


136 


158 


2 30 


3 21 


4 59 


5 


1'15 


1'31 


151 


3 20 


3 05 


4 25 


7 34 


25 18 





For Latitude 42° 30'. 



Oh. 


24" 


30" 


36" 


44" 


52" 


1'02" 


1'14" 


1'29" 


1'49" 


2 


28 


35 


39 


50 


I'OO 


1 12 


126 


1 45 


2 11 


3 


36 


43 


52 


102 


113 


129 


149 


2 17 


2 59 


4 


50 


I'OO 


I'll 


126 


144 


2 10 


2 49 


3 55 


6 16 


5 


ri6 


136 


158 


2 30 


3 22 


5 00 


9 24 







For Latitude 45°. 



Oh. 


27" 


33" 


40" 


48" 


57" 


1'08" 


1'21" 


1'39" 


2 


32 


39 


46 


52 


1'06 


1 19 


135 


157 


3 


40 


47 


56 


1'07 


121 


138 


2 00 


2 34 


4 


54 


1'04 


1'16 


133 


154 


2 24 


3 11 


4 38 


5 


1'23 


141 


2 05 


2 41 


3 40 


5 40 


12 02 





2'02" 

2 29 

3 29 
815 



For Latitude 47° 30' 



Oh 


30" 


36" 


44" 


52" 


1'02" 


1'14" 


1'29" 


1'49" 


2 


35 


42 


50 


I'OO 


112 


1 26 


145 


2 01 


3 


43 


51 


I'Ol 


113 


128 


147 


2 15 


2 56 


4 


56 


1'09 


123 


140 


2 05 


2 40 


3 39 


5 37 


5 


1'27 


1 46 


2 12 


2 53 


4 01 


6 30 


16 19 





2'18" 

2 51 

4 08 

1118 



For Latitude 50°, 



Oh. 

2 

3 

4 

5 



33" 

38 

47 

1'03 

130 



40' 

46 

56 

1'44 

151 



48" 

55 

1'06 

129 

2 19 



57' 
1'06 
119 
148 
3 04 



1'08" 
118 
136 
2 16 
4 22 



1'21" 
135 
2 29 

2 58 
7 28 



VSd" 
157 
2 31 
4 18 
2410 



2'02" 

2 28 

3 23 
6 59 



2'36" 
3 19 

5 02 
19 47 











For Latitude 52 


°30'. 








Oh. 


36" 


44" 


52" 


1'02" 


1'14" 


1'29" 


1'49" 


218" 


305" 




2 


43 


50 


59 


111 


126 


142 


2 23 


2 49 


3 55 




3 


50 


I'OO 


I'll 


126 


145 


2 11 


2 51 


2 58 


6 22 




4 


1'05 


118 


135 


2 10 


2 28 


319 


4 53 


8 42 






5 


134 


156 


2 27 


3 16 


4 47 


8 52 









12 



ELEMENTS OF SURVEYING. 



[a PP. A. 



Oh. 

2 

.3 

4 

5 



DECLINATIONS. 



For Latitude 55' 



+ 20° 


+ 15° 


+ 10° 


+ 5° 


0° 


— 5° 


— 10^ 


— 15° 


40' 


48" 


57" 


I'OS" 


1'21" 


1'39" 


2' 02" 


2'36" 


46 


55 


1'05 


1 18 


1 34 


156 


2 80 


3 15 


55 


106 


1 19 


1 35 


1 58 


2 80 


8 21 


4 58 


I'lO 


1 23 


142 


2 06 


2 43 


344 


5 49 


12 41 


137 


2 01 


2 84 


3 28 


5 15 


10 18 







-20^ 

3'33' 
4 47 
9 19 



For Latitude 57 30' 



Oh. 


44" 


52" 


1'02" 


1'14" 


1'29" 


1'49" 


2'18" 


305' 


2 


50 


59 


1 11 


1 25 


143 


2 09 


2 47 


3 51 


3 


58 


I'lO 


1 24 


142 


2 07 


2 43 


3 45 


5 50 


4 


I'll 


1 25 


1 43 


2 10 


2 50 


8 55 


6 14 


14 49 


5 


141 


2 06 


2 42 


342 


5 46 


13 26 







4'37" 
6 04 

12 47 



Explanation of the Table of Refractions.— The table is cal- 

c'.ilated for latitudes between 30 "" and 50' at intervals of 2^°, that being as near 
a!3 is required. 

The declination ranges from to 20° both north and south, the + declina- 
tions beincr north, and — south, and is given for every five degrees, that 
being sufficiently near for all practical purposes. 

The hour angle in the first column indicates the distance of the sun from 
the meridian in hours, the refraction given for hours being that which 
aflfects the observed declination of the sun when on the meridian, commonly 
known as meridional refraction ; the refraction for the hours just before and 
after noon is so nearly that of the meridian, that it may be called and allowed 
as the same. 

When the table is used, it must be borne in mind that when the declina- 
tion is north or -f in the table, the refraction is to be added ; when the decli- 
nation is south or — , the refraction must be subtracted. 

It will be noticed that the refraction in south or — declination increases 
very rapidly as the sun nears the horizon, showing that observations should 
not be taken with the sun when south of the equator, less than one hour 
fr<tm the horizon. 

The calculation of the declination for the different hours of the day should 
of course be made and noted before the surveyor commences his work, that 
he may lay off the change from hf iir to hour, from a table prepared before- 
hand. 



A pp. A.] 



THE SOLAR COMPASS. 



13 



Solar Attachment to Transit. 

The Solar Attachment is CBsentially the solar ajjparatus of Burt placed 
upon the cro88-bar of the ordinary transit, the i>o]ar axis only being directed 
above instead of below, as in the solar compass. 

A little circular disc of an inch and a half in diameter, and having a short 
round pivot projecting above its uf>per surface, is first securely screwed to 
the telescope axis. 

Upon this pivot rests the enlarged base of the polar axis, which is also 
firmly connected with the disc by four capstan head screws passing from the 
under side of the disc into the base already named. 

These screws serve to adjust the polar axis. 

The hour circle surrounding the 
base of the polar axis is easily mov- 
able about it, and can be fastened 
at any point desired by two flat- 
head screws above. It is divided 
to five minutes of time ; is figured 
from I. to XII., and is read by a 
email index fixed to the declination 
circle, and moving with it. 

A hollow cone, or socket, fitting 
closely to the polar axis and made 
to move snugly upon it, or clamped 
at any point desired by a milled- 
head screw on top, furnishes, by 
its two expanded arms below, a 
firm support for the declination 
arc, which is securely fastened to 
it by two large screws, as shown. 

The declination arc is of about 
five inches radius, is divided to 
quarter degrees, and reads by its 
vernier to single minutes of arc, 
the divisions of both vernier and 
limb being in the same plane. 

The declination arm has the 
usual lenses and silver plates on 
the two opposite blocks, made pre- 
cisely like those of the ordinary 
solar compass, but its vernier ^is 
outside the block, and more easily read. 

The declination arm has also a clamp and tangent movement, as shown 
in the cut. The arc of the declination limb is turned on its axis and one or 
the other solar lens used, as the sun is north or south of the equator ; the cut 
above shows its position when it is north. 

The latitude is set off by means of a large vertical limb having a radius of 
two and a half inches ; the arc is divided to twenty minutes, is figured from 
the centre, each way, up to 80% and is read by its vernier to single minutes. 




PACV*** 



Fig. 3, 
Showing a Transit with Solar Attachment. 



14 ELEMEin^S OP SURVEYING. [aPP. A. 

It has also a clamp-screw inserted near its centre, by which it can be set 
fast to the telescope axis in any desired position. 

The vernier of the vertical limb is made movable by the tangent-screw 
attached, so that its zero and that of the limb are readily made to coincide 
when, in adjusting the limb to the level of the telescope, the arc is clamped 
to the axis. 

The usual tangent movement to the telescope axis serves, of course, to 
bring the vertical limb to the proper elevation, as hereafter described. 

A level on the under side of the telescope, with ground vial and scale, is 
indispensable in the use of the Solar attachment. 

To Find the Latitude — First level the instrument very carefully 
using the level of the telescope until the bubble will remain in the centre 
during a complete revolution of the instrument, the tangent movement of the 
telescope being used in connection with the leveling screws of the parallel 
plates, and the axis of the telescope firmly clamped. 

Next clamp the vertical arc so that its zero and that of its vernier coincide 
as near as may be, and then bring them into exact line by the tangent-screw 
of the vernier. 

Then, having the declination of the sun for 12 o'clock of the given day as 
affected by the meridional refraction carefully set off upon the declination arc, 
note also the equation of time and fifteen or twenty minutes before noon, the 
telescope being directed to the north, and the object end lowered until, by 
moving the instrument upon its spindle and the declination arc from side to 
side, the sun's image is brought nearly into position between the equatorial 
lines. Now bring the declination arc directly in line with the telescope, 
clamp the axis firmly, and with the tangent screw bring the image precisely 
between the lines and keep it there with the tangent screw, raising it as long 
as it runs below the lower equatorial line, or, in other words, as long as the 
sun continues to rise in the heavens. 

When the sun reaches the meridian the image will remain stationary for 
an instant and then begin to rise on the plate. 

The moment the image ceases to run below is of course apparent noon, 
when the index of the hour arc should indicate XII., and the latitude be deter- 
mined by the reading of the vertical arc. 

It must be remembered, however, that the angle through which the polar 
axis has moved in the operation just described is measured from the zenith 
instead of the horizon as in the ordinary solar, so that the angle read on the 
vertical limb is the complement of the latitude. 

The latitude itself is readily found by subtracting this angle from 90° ; 
thus at Troy, the reading of the limb being found as above directed to be 
47° 16', the latitude will be 90° -47° 16' =42° 44'. 

It will be noticed that with this apparatus the latitude of any place can 
be most easily ascertained without any index error, as in the usual solar 
compass. 

To run lines with the Solar Attachment — Having set off 

the complement of the latitude of the place on the vertical arc, and the decli- 
nation for the given day and hour as in the solar, the instrument being also 
carefully leveled by the telescope bubble, set the horizontal limb at zero and 



APP. A.] THE SOLAR COMPASS. 15 

clamp the plates together, loosen the lower clamp so that the transit moves 
easily upon its lower socket, set the instrument approximately north and 
south, the object end of the telescope pointing to the north, turn the proper 
solar lens to the sun, and with one hand on the plates and the other on the 
revolving arm, move them from side to side until the sun's image is brought 
between the equatorial lines on the silver plate. 

The lower clamp of the instrument should now be fastened, and any fur- 
ther lateral movement be made by the tangent screw of the tripod. The 
necessary allowance being mado for refraction, the telescope will be in the 
true meridian, and being undamped, may be used like the sights of the ordi- 
nary solar compass, but with far greater accuracy and satisfaction in estab- 
lishing meridian lines. Of course when the upper or vernier plate is un- 
damped from the limb, any angle read by the verniers is an angle from 
the meridian, and thus parallels of latitude or any other angles from the true 
meridian may be established as with the solar compass. 

The bearing of the needle, when the telescope is on the meridian, will also 
give the variation of the needle at the point of observation. 

If the instrument has a movable compass circle, the variation of the 
needle can be set off to single minutes, the needle kept at zero, or " with the 
sun," and thus lines be run by the needle alone when the sun is obscured. 



APPENDIX B. 



THE SEXTANT.* 

(By Prof. J. K. Rees, Columbia College.) 

This instrument is especially useful to tbe scientific exiJorer on account 
of its portability and siniplicity of manipulation. It requires no fixed su])- 
port, and furnishes data with the least expenditure of the time of the ob- 
server. The accuracy of fixed instruments is not to be expected from it, 
since it is lu^ld in the hand and is of small dimensions. 

The Principle of the Instrument.— The optical principle upon 

which the sextant is made 
is : — If a ray of light suffers 
two successive letiections in 
the same plane by two plane 
mirrors, the angle between 
the first and last directions 
of the ray is twice the angle 
of the mirrors. 

Let / and H be two 
plane mirrors perpendicular 
to the plane of the paper, — 
which is taken as the plane 
of reflection. 

A ray of light from A is 
reflected, first from the mir- 
ror 1 in the direction GO, 
then by the mirror fi" along 
OT. The angle between 
the first and last direction 
of the ray after these two 
reflections is ATO. 

Draw GN and OM nor- 
mal to the mirrors / and H 
respectively. Then NKO 
equals the angle of the mirrors. 

From the law of the reflection of light it is known that the angle 

AGN oxi = tLTigleNGO; 
and also, GOMot i' = MOT. 




Fig. 4. 



♦ For a fiill description of this instrument, see Chauvenet'e Spherical and Practical 
Aetronomy, published by Lippiucott io 1863, 



APP. B.] 



THE SEXTANT. 



17 



ce angle 
angle 



A10 = 2i - 2i' ; 
NKO = i - i' ; 
ATO = 2NK0. 



Q. E. D. 



II 



Suppose now that the glass His unsilvered on the upper half; then a ray 
of light coming from B will pass through this unsilvered portion to T, and 
the angle ATO will measure the angniar distance of ^ from B as seen at T. 

To apply this principle the mirror /, revolving about a pivot at C, has at- 
tached to it an arm or bar II), which, as the mirror is turned, moves over a 
graduated arc BR. The mirror H is fixed in position. There will be one 
position of the index arm where the two mirrors will be parallel. Then since 
the angle bet wean the first and last directions of the ray of light which is 
reflected by both mirrors is zero, or the two directions are parallel, the indi- 
cated point of the graduated arc is marked zero. The graduations are then 
continued to the left, calling each degree two degrees, in order to read off at 
once the required angle. 

The best form of the common sextant is seen in the accompanying cut, 
furnished by Messrs. Stackpole & Brother, N. Y. : 

The frame is of brass » 
constructed so as to com- 
bine strength with light- 
ness ; the graduated arc, 
inlaid in the brass, is usu- 
ally of silver. The divi- 
sions of the arc are ordi- 
narily 10' each, which are 
subdivided by the vernier 
to 10". The handle A by 
which it is held in the hand 
is of wood. 

The mirrors / and H 
are of plate glass silvered. 
The upper half of the glass 
H is left unsilvered in or- 
der that the direct rays 
from a distant object may not be interrupted. To give greater distinctness 
to the images a small telescope T is placed in the line of sight OT. The 
telescope is supported in a ring B, which can be moved in a direction at 
right angles to the plane of the sextant. Thus the axis of the telescope can 
be directed either towards the silvered or the unsilvered part of the mirror. 
This motion changes the plane of reflection, which, however, remains 
always parallel to the plane of the sextant ; the use of the motion being 
merely to regulate the relative brightness of the direct and reflected images. 
The vernier is read with the aid of a glass G which is attached to the index 
bar. The central mirror 1, or index glass, is fastened in a brass frame which 
is firmly attached to the index bar by three screws. This glass is generally 
set by the maker so as to be perpendicular to the plane of the sextant. There 
are no adjusting screws usually connected with it. The fixed mirror E, or 
horizon glass (so called because through it the horizon is observed in taking 




Fig. 5. 



18 ELEMENTS OF SURVEYING. [APP. B. 

altitudes), is specially provided with screws bv which its position with respect 
to the sextant plane may be rectified. 

At P and Q are colored glasses of different shades, which may be used 
separately or in combination, to defend the eye from the intense light of the 

SOB. 

Common Adjustments of Ordinary Sextant. 

1. The Index Glass must he perpendicular to the plane of the sextant. 

2. The Horizon Glass must also be perpendicular to the plane of the sextant. 

3. The central sight line of the telescope must be parallel to the plane of the 
sextant. 

4. The true zero of the arc must befcnmd. 

1. AdjUStmeilt of the Index Glass.— Bring the vernier to 
about the middle of the graduated arc ; then, placing the eye a little above 
the plane of the sextant and near the index glass, examine the direct and re- 
flected images of the graduated arc. If the one appears to run into the other 
the index glass is perpendicular to the plane of the sextant, and the adjust- 
ment is complete. 

If the reflected image appears too high or too low the glass leans forward 
or backward. The glass may then be adjusted to perpendicularity by placing 
a piece of paper under one edge of the plate by which the glass is held to 
the index arm, first loosening the screws ; or the glass may be taken out of 
the frame, and the supports against which the glass leans may be filed so as 
to bring the glass, when set back, perpendicularly to the plane of the sextant. 

2. Adjustment of the Horizon G-lass.— This glass must also 

be perpendicular to the plane of the sextant. The index glass having been 
adjusted to perpendicularity, if it is fotmd that in any one position the hori- 
zon glass is parallel to the index glass, then the hori2on glass is perpendicular 
to the plane of the sextant. 

In order to test this parallelism, put in the telescope and direct it to a star 
or any distant, well-defined terrestrial object. Move the index bar until the 
direct and reflected images are in the field of view, then clamp the vernier, 
and by moving the tangent screw cause one image to pass the other ; if they 
pass exactly one over the other the adjustment is complete ; if they pass one 
at the side of the other, the horizon glass must be adjusted. There are ad- 
justing screws attached to the glass whereby it can be inclined to or from the 
sextant plane, and also turned around an axis perpendicular to the sextant 
plane. By means of the first set of screws the adjustment for perpendicu- 
larity can be made, and by means of the second set the position of the zero of 
the limb can be altered to a small extent. 

3. Adjustment of the Telescope.— The sight line of the tele- 
scope is the line from the centre of the field of viewtlirough the centre of the 
object glass. This line must be parallel to the plane of the sextant. In 
order to test for this, choose two distant objects like the sun and moon, 90' to 
120' apart ; direct the telescope to one of these objects, holding the plane of 
the sextant so as to pass through both ; then moving the index bar, bring 
the second object into the field of view ; clamp the vernier, turn the tangrent 



APP. B.] THE SEXTANT. 19 

screw until the two objects are tangent to each other on the thread of 
the telescope nearest to the instrument. Then by moving the instrument, 
cause the objects to come on tlie thread farthest from the instrument. If the 
tangency is still perfect, the adjustment is complete. If the objects separate 
upon the thread farthest from the instrument, tiien the object end of the 
telescope droops towards the plane of the instrument ; if the images overlap, 
then the telescope inclines upward from the plane of the instrument. The 
adjustment is made by means of the screws that work into the collar which 
carries the telescope. 

4. Index Correction.— When the two glasses are parallel the zero 
of the graduated limb should coincide with the zero of the vernier. This 
adjustment must be very carefully looked after before taking any observa- 
tions, because it is an adjustment that is liable to change. Rather than make 
this adjustment accurately every time an observation is made it is prefer- 
able to determine the place of the true zero of the graduated arc and allow 
for the correction. This correction is simply the distance between the gradu- 
ated zero of the instrument and the reading of the vernier when the two 
mirrors are parallel. This correction is minus when it is on the graduated 
arc towards the increasing numbers of the graduation, and plus when on the 
opposite side. 

The index correction may be determined by observations upon a star, or a 
distant, well-defined, terrestrial object, or upon the sun. 

First, by a star : Direct the telescope to a star of the third or fourth mag- 
nitude ; move the index bar until the reflected image of this star comes into 
the field of view ; then clamp the vernier ; turn the tangent screw until the 
direct and reflected images of the star are exactly in coincidence ; take the 
reading of the vernier; apply the proper sign, and the arc reading is the 
index correction. 

Second, by a distant, well-defined terrestrial object, or the reflection of 
sunlight from — for example — the bulb of a thermometer or a drop of water. 
This can be observed in the same way as a star, although not giving as accu- 
rate results. 

Third, by the sun : Turn on the colored glasses until the light from the 
sun is diminished sufficiently to suit the eye ; bring the direct and reflected 
images of the sun into the field of view ; clamp the vernier ; turn the tangent 
screw until one image of the sun is tangent to the other ; take the reading ; 
turn the tangent screw until the contact is broken ; bring the images back to 
tangency ; take the reading again ; in this way make five readings : then turn 
the tangent screw until the images change places and tangency is made on 
the other side ; take the same number of readings here. 

In order to read always from the same end of the vernier, call the zero of 
the vernier 360°, and read the vernier accordingly. Take tbe mean of the 
readings in the first and second cases, add them together, and divide by two. 
Subtract the result from 360°, and the difierence will be the index correction. 

In order to avoid the effect of refraction it is best to measure the horizontal 
diameter of the sun. To check the observations compute the diameter of the 
sun from data given in the Nautical Almanac, for the day of the observation, 
and compare it with the diameter of the sun as obtained from the obser- 
vations. 



20 ELEMEl^TS OF SUKVEYIN^G. [aPP. B. 

Note.— Let E be tlie true reading of the vernier when the mirrors are 
parallel; let 8 be the diameter of the sun ; let r be the reading of the vernier 
when contact is made on the left of the zero of the instrument, and r' the 
reading when the contact is made on the right. 

Then, r = R + 8 

r' = B- 8. 



Hence, R — 

and 8 = 



r + r' 
r — r' 



June 1st, 1883.— 10 a. m. 



Observed sun for index correction. 



ON 


ARC. 


OFF 


ARC. 


360° 34' 


10" 
10" 

5 

10 



359 


30 


50 
45 
45 
55 
50 


360 34 


07 = r 


means 359 


30 


49 = r' 


359 30 


49 


360 


34 


07 


2)720 4 


56 


2) 1 


3 


18 


360 2 


28 


Observed diam. sun, 


31 


39. 


Index correction. 




Calculated diam. sun, 


31 


36.7 


-2' 28" 


Difiference, . . . 


, 


3.3 



To Measure the Angular Distance between two objects with the 

Sextant. 

Turn the eye-piece until two of the reticule threads of the telescope are 
parallel to the plane of the instrument ; then direct the telescope to the fainter 
of the two objects ; move the plane of the instrument until it passes through 
the two objects ; revolve the index bar until the reflected image of the second 
object appears in the field of view ; clamp the vernier ; with the tangent 
screw make the two images coincide. The reading of the vernier with the 
index correction applied will be the angular distance of the two objects. The 
index glass must always be on the side towards the second object. The coin- 
cidence must be made at the middle point of the field of view. Care must 
be taken to have the relative brightness of the two objects about the same. 

Altitudes of objects may be obtained in a similar way, holding the plane 
of the sextant vertical. 



APPENDIX C. 



INSTRUCTIONS TO DEPUTY U. S. MINERAL 
SURVEYORS. 

Surveyor General's Office, 

Denver, Colorado, June 1, 1880. 

Sir : — The following instructions have been prepared for your government 
in making mineral surveys : 

1. You can make no official survey without an order from this office. 

2. To procure an order for survey, tlie application should be made in 
writing, accompanied with a copy of the certificate of location of the claim to 
be surveyed, duly certified by the proper recorder, and inclosing a deposit of 
$35 for office work. Be careful to state plainly the name of the person or 
persons who desire the survey. 

3. An order will not be issued unless in the judgment of this office the 
certificate of location is in accordance with law and regulations. 

4. For the accommodation of claimants residing remote from a United 
States Depository, this office will see that the deposit is made when the 
proper funds are received. Send Post Office order, draft or check, on Denver, 
as other drafts cannot be used without discount. 

5. Should the original certificate of location be found defective, and the 
description does not cover the ground claimed, an additional or amended cer- 
tificate may be filed as provided by section 13 of the laws of Colorado, ap- 
proved February 13th, 1874, and a certified copy of the amended certificate 
must be sent to this office, and your survey dated subsequent to the filing of 
the same in the proper recorder's office. 

6. Certificates of location are frequently sent, which any well-informed 
deputy should know are too indefinite ; examine them carefully before send- 
ing, and it will save yourselves and this office much trouble, and remember 
that your survey MUST agree with certificate of location. 

7. "If the records of locations made prior to the passage of the mining 
act of May 10th, 1872, are not sufficiently definite and certain to enable you 
to make a correct survey therefrom, you should, after reasonable notice in 
writing, to be served personally, or through the United States mail, on the 
applicant for survey and adjoining claimants (whose residence or Post Office 
address you may know or can ascertain by the exercise of reasonable dili- 
gence), take the testimony of neighboring claimants and other persons who 
are familiar with the boundaries thereof as originally located and asserted by 
the locators of the claim, and after having ascertained by such testimony the 



22 ELEME^-TS OF SUEVEYING. [aPP. C. 

boundaries as originally established, you should make a survey in accordance 
therewith, and transmit full and correct returns of survey, accompanied by 
the copy of the record of location, the testimony, and a copy of the notice 
served on the claimant and adjoining proprietors, certifying thereon when, in 
what manner, and on whom, service was made." 

8. The act of Congress of May 10th, 1872, expressly provides that " the 
location must be distinctly marked on the ground, so that its boundaries can 
be readily traced," and " that all records of mining claims hereafter made 
shall contain the name or names of the locators, the date of location, and 
such a description of the claim or claims, located by reference to some natural 
object or permanent monument, as will identity the claim." 

9. " These provisions of the law must be strictly complied with in each 
case to entitle a claimant to a survey and patent, and therefore should a claim- 
ant under a location made subsequent to the passage of the mining act of May 
10th, 1873, who has not complied with said requirements in regard to mark- 
ing the location upon the ground, and recording the same, apply for a survey," 
I ''will decline to order it." 

10. " The only relief for a party under such circumstances will be to make 
a new location in conformity to law and regulations, as no case will be ap- 
proved by this office unless these and all other provisions of law are substan- 
tially complied with." 

11. Comers must be established in as permanent a manner as possible, 
and should consist of rock in place, or tree if they are found at the exact point ; 
otherwise a STONE, not less than two feet long, set one foot in the ground ; 
or POSTS, not less than four inches in diameter, planted two feet in the ground 
and protruding not less than two and not more than four feet above ground ; 
both stones and posts must be protected with a mound of stone or earth in 
addition to the planting. 

12. Each corner to be marked No. 1, 3, etc., also the number of the sur- 
vey, as you proceed with the work, giving in field notes, bearings and distances 
from each corner to rocks or trees, if any such are at convenient distance, 
marking same wdth number of corner and number of survey, and describe 
marks. Wooden posts and trees must be marked with a scribe, and rocks 
with a chisel. Give, from at least two corners of the survey, two or more 
bearings to well known points, such as mountain peaks, confluence of streams, 
etc., both in field notes and on plats. If any portion of this section cannot 
be complied with, so state it in field notes. 

13. Note all objects crossed by your line of survey, such as previous sur- 
veys, lodes, ditches, roads, ravines, etc. , etc. , and show them upon plat. " If 
in running the exterior boundaries of a claim it is found that two surveys 
conflict, the plats and field notes should show the extent of the conflict, giv- 
ing the area embraced in both surveys, and also the courses and distances 
from the established corners at which the exterior boundaries of the respec- 
tive surveys intersect each other. In notes give area as follows : 

Total area, 5. 94 acres. 

Less area in conflict with surveys Nos. 13 and 17, . . 3.00 acres. 



Leaving net area, 3.94 acres. 

On plat give net area only. 



APP. c] i:n'structions to u. s. mineral surveyors. 23 

14. Connect corner No. 1 of your survey with some corner of the public 
survey if the claim is located within two miles of such public survey. From 
corner No. 1 beginning you will proceed with the survey of the claim, giving 
distances in feet and true courses, as you proceed establishing a corner at each 
angle of the survey. If an official survey bas been made in the same vicinity, 
run a connecting line to some corner of the same. Surveys must be made to 
close exactly. Mention particularly all adjoining claimants. 

15. In referring to other surveys give in your field notes the name and 
number of each survey, and name of claimant or owner. When referring to 
a survey more than once it is unnecessary to repeat more than the number. 

16. See Section 2320, Revised Statutes of the United States, in regard to 
width of lode claims " on each side of the middle of the vein at the surface." 
When the locator does not determine, by exploration, where the middle of the 
vein at the surface is, his discovery shaft must be assumed to mark such 
point." 

17. You will give the quarter section, township and range in which the 
claim is situated, in notes and on plat, showing section lines on your plat in 
black, and quarter section lines in red. 

18. In districts where there are no public surveys within two miles of 
your survey, and no locating monument previously established within one 
hundred chains of such survey, you will establish a LOCATING MONU- 
MENT at some well-known point, a rock in place being preferable ; but if 
such cannot be found, then erect a large substantial mound of rocks, describing 
the same fully in your notes, and hereafter chisel upon it a name, using this 
as a starting point for your surveys, and when the public surveys reach the 
locality, run a connecting line from the monument to a corner in said surveys, 
thus connecting all claims surveyed from the monument. 

19. Note all improvements upon the claim, such as shafts, drifts^ adits, 
cuts, buildings, etc. , giving the extent of same. Show improvements on plat 
and locate them in field notes, by course and distance, in a direct line from 
some corner of the survey. 

20. After describing fully the improvements (stating if the excavations 
are in dirt or rock) placed on the claim by the claimant or his grantors, say 
that " the value of the said improvements, together with the labor expended 
on the said claim by the claimant (or his grantors, as the case may be) is not 
less than five hundred dollars." If $500 in improvements has not been ex 
pended at time of survey the work maybe performed and certificate of Sur- 
veyor-General filed at the land office during the 60 days of advertising. To 
obtain this certificate the deputy must send his affidavit stating that the sum 
of $500 has been expended, describing improvements fully. In estimating 
the value of improvements only actual mining improvements should be con- 
sidered. 

21. " From an examination of the returns of surveys of mining claims, I 
am satisfied that in many instances the deputy surveyors certify to the value 
of improvements without ascertaining whether such improvements are made 
by the claimant or his grantors, or other persons." 

22. " No improvements should be included in the estimate unless they 
have been made by the applicant for survey, or by those from whom he de- 
rives his title, and so stated in your notes." 

23. " The value of improvements made upon other locations, or by other 



24 ELEMENTS OF SUEVEYI^S'G. [APP. C. 

persons, should not be taken into consideration, but excluded by deputies in 
their'estimate of improvements upon tlie claim." 

24. Following description of improvements made by the claimant, locate 
the improvements made by other persons, in the manner described in Sec. 19. 

25. State all facts coming to your knowledge in regard to adjoining 
conflicting claimants, whether their claims are surveyed or not. 

26. Your field notes should be complete in themselves, leaving nothing 
to be explained by letter ; and as they are bound in book form after approval 
you shoidd leave margin for binding. Fold notes so as to leave as little 
blank paper as possible, and stitch them together. 

27. If you know your survey does not agree with that of other deputies 
you should communicate with such deputy before sending your survey to 
this ofiice, and try to reconcile all discrepancies ; if it cannot be adjusted, re- 
port to this office, and a joint survey will be ordered upon knowing that this 
section has been complied with. 

28. When a joint survey is ordered, the deputy who discovers the sup- 
posed error will be directed to call upon the deputy supposed to be in error, 
and see if the discrepancy cannot be reconciled. If they cannot agree, then 
they shall make a joint survey within ten days after the date of the order. 
When said survey is completed they shall make a joint affidavit to the field 
notes, and forward them to this office. The deputy found to be in error shall 
pay all expenses, including $10 per day to the deputy whose work is found 
to be correct. If both deputies are found to be in such error as to require 
amended field notes of their former survey, then each deputy shall pay one- 
half the expenses of the joint survey. Any deputy refusing or neglecting to 
appear for the joint survey, within the ten days named, or who shall refuse 
or neglect to pay the expenses as above indicated, will be suspended ; and if 
the refusal or neglect shall extend to twenty days, his commission as deputy 
will be revoked. 

29. In preparing plats make the top North, and color only the ground not 
in conflict. Mark upon the plat all corners (thus, " Cor. No. 1," etc.), courses 
to mountain peaks, courses and distances to all connecting lines and upon the 
boundaries of the survey. Plat survey upon as large a scale as paper will 
admit of, if practicable not less than 200 feet to the inch. Be careful that 
your plats and notes agree. 

30. An applicant has the right to abandon, from his application for patent, 
any part or portion of the premises embraced in the survey, but in case he 
does abandon any portion of the premises embraced by his application and 
survey, it will be necessary that an amended survey be made upon applica- 
tion of the claimant, and as such amended survey involves as much office work 
as the original survey, the usual deposit should accompany the claimant's 
application for the amended survey. 

31. After this date surveys will be approved in the order in which they 
are received, deputies being required to act promptly in all official matters, 
that the approval of a survey may not be unnecessarily delayed. In case of 
unreasonable delay on the part of the deputy to correct errors, the survey will 
be stricken from the files, and the applicant notified. 

32. In order that all returns may be made uniform, blank plats and field 
note paper wiU be furnished from this office for your use, which blanks you 
wiU fill up as the case may require. You will be careful that the names of 



APP. C] IKSTRUCTIONS TO U. S. MII^ERAL SUEVEYORS. 25 

claimants and number of survey agree with the order for survey and certifi- 
cate of location! You will make one copy of the plat and notes and transmit 
the same to this oflBce, prepaying the postage or express, in full ; otherwise 
they will not be received. Do not fold plats for transmission, but roll them. 
As your plat and notes come under the head of •' written matter," they 
require letter postage. 

33. Unless plats and notes are prepared in a neat and workmanlike man 
ner they will not be accepted. Plats and notes are often found to be incor- 
rect from negligence, carelessness and ignorance. Many deputies appear to 
depend upon this oflBce to detect and correct these errors, as it saves them the 
trouble. Hereafter such surveys will be returned with the simple statement 
that " they are incorrect." 

34 A solar transit must be used in all official surveys being guided by 
the solar apparatus, and not the needle, unless the courses are deflected from 
a meridian astronomically established. On account of local attractions a 
needle instrument will not be accepted as reliable. State in field notes the 
kind of instrument used, and the manner in which the courses are taken. 

35. " You are informed that the employment of Deputy Mineral Survey- 
ors as attorneys in mineral claims, directly or indirectly, is absolutely prohib- 
ited," and you will make no survey in which yoa are interested. 

36. Be particular to write all proper names plainly. 

37. Deputies changing their residence should notify this office of such 
change. 

38. I shall expect you to make yourself thoroughly familiar with all the 
mining laws, National and State, as well as with these instructions, and I am 
sure that errors will be less frequent, and this office as well as yourself will 
be relieved of much annoyance. 

39. All official communications must be addressed to the Surveyor Gen- 
eral, and not to clerks in his office. 

Very respectfully, 

Albert Johnso?^, 

Surveyor General of Colorado. 



Sample Field Notes furnished Deputies. 
Survey No. 500, District No. 3. 

FIELD NOTES 

Of the survey of the claim of 

... .John H. Marshall, 

upon the Excelsior Lode, 

in. . . . California Mining District, 

Lake Coimty, Colorado. 



Surveyed by ...... . George D. Williams, ; 

U. S. Deputy Mineral Surveyor. 

Survey begun October 23d, 1881, 

and completed > " 38th, .1881, 



u 



ELEMEirrS OF SUEVEYnrO. 



[app. c. 



Fee(. 



DEscEIPTIO^' OF Survey. 
Beginning at Cor. Xo. 1, a spruce post, 5 ft. long, 4 ins. 
diam., set 2 ft. in the ground, with mound of stone, 
marked 5 Jo whence the W. 34, cor. in sec. 16, T. 8 S., 
R. 80 W. of the 6th Prin. Mer. bears, 

S. 84= 2Q' W. 1847. 4 ft. 
A spruce, 6 ins. diam., marked B. T. gJo, bears S. 78° 32' 
W. 32.9 ft. 

Thence N. 11° 53' E. 
Vap. 14= 15' E. 

300. To Cor. No. 2. 

A spruce post 4 ft. long. 4 ins. dia., set 2 ft. in ground, 
with mound of earth marked -Iq, whence, 
Holr Cross Mt. bears X. 41-37' W. ; 
Mt. Leon bears X. 14= 29' W. 
(Xo other bearings available.) 

Thence S. 78^ 7' E. 
Var. 14= 15' E. 

To road running S. Easterly. 

224.27 Intersect line 4-1 of Sur. No. 495, Treasury lode, John 
P. Jones et al. claimants, N. 15= E. 79.91 ft. from Cor. 
No. 4. 

952.72 Intersect line 4-1 of Sur. Xo. 227, Silver Dollar Lode, 
Henrv J. Smith, claimant, at S. 3= W. 401.02 ft. from Cor. 
No. 1. 

1256.36 Intersect line 2-3 of Sur. No. 227, S. 3' W. 447 9 ft. from 
Cor. No. 3. 



1500. 



To Cor. No. 3. 



APP. C] 



SAMPLE FIELD KOTES. 



27 



Feet. 



149.5 

198. 
300. 



112. 
196.75 

500.39 

911. 
1500. 



A granite stone 25 x 11 x 7 ins., set 1 ft. in ground, 
with mound of stones, chiseled 5^0, whence a spruce 
6 ins. dia., marked B. T-. ^%^, bears S. 80" 23' E. 20.4 ft.; 
B. R. 5^0 chiseled on rock in place, bears N. 44° E. 

30 ft. 

Thence S. 11° 53' W. 
Var. 14° 55' E. 

Intersect line 3-4 of Sur. No 495, at N. 75° W., 218 ft. 
from Cor. No. 3. 

To G-ulch ; course N. E. 

To Cor. No. 4. 

A quartz stone 24 x 13 x 8 Ins., set 1 ft. in ground, and 
mound of stone, chiseled ^%^, whence B. R. g^^ chiseled 
on large boulder, bears N. 88° 22' W. 25.4 ft. 

Mt. Leon bears N. 19° 22' W. 
Sherman Peak bears S 84° 45' E. 
Cor. No 3, Sur. No. 495, bears N. 69° 23' E. 258 ft. 
Cor. No. 3, Sur. 227, bears S. 18° 9' W. 743.9 ft. 

Thence N. 78° 7' W. 
Var. 14° 55' E. 
To Gulch. 

Intersect line 2-3 of Sur. No. 227, N. 3° E., 748.45 ft. from 
Cor. No. 3. 

Intersect line 4-1 of Sur. No. 227, N. 3° E., 795.34 ft. from 
Cor. No. 4. 

To road. 

To Cor, No. 1, place of beginning. 



28 ELEMElfTS OF SURVEYING. [APP. a 

Akea. 

Containing 10.32 acres. 

Less conflicts with .... 

Sur. No. 227=2.13 acres 

" " 495=2.46 " 4.58 " 



acres. 



Leaving net area 5.74 

Location. 

This survey is located in the N. 3.^ and S. E. U of Sec. 16. in T. 8 S R 
SOW. 

Impkovements 
Upon this claim consist of : 

A discovery shaft, 5' x 3', 275 ft. deep, timbered, which bears from Cor 
No. 1, N. 64° 59' E. 250.6 ft. 

A log shaft-house, 56' x 40', the S. W. corner of which bears from Cor 

No. 1, N. 65= 26' E. 207 ft. 

A shaft 4' X 4', 25 ft. deep, which bears from Cor. No. 1, N. 88° 39 E. 
765 ft. 

An open cut 3' x 10', which bears from Cor. No. 3, S. 58° 45' W. 264 ft. 

JL hereby certify that the value of said improvements, together with the 
labor expended on the said claim by the claimant and his grantors, is not 
less than Five Hundred Dollars ($500). 

Improvements made by other parties are as follows : 

A shaft 4' x 6', 78 feet deep, which bears from Cor. No. 3, N. 83° 37' W. 
389 ft. 

Adjoining Claims. 
Surveys Nos. 495 and 227. No others known. 

Instbuments Used. 

A Gurley Mountain Transit, with solar attachment (or the instrument 
used), and steel tape (or the measure used). 

All courses determined by the use of the solar apparatus. 

Address of Applicant, 

JOHN H. MARSHALL, 

P. O. Box 743, Leadville, Colo. 

[Tlien follow a list of the assistants employed in making the survey, with 
their aflBdavit as to its conformity to instructions, the affidavit of the Deputy 
Mineral Surveyor, and the certificate of the Surveyor General.] 



A pp. C] 



SAMPLE PLAT. 



29 




9/ &B.!iS«XKT8iS.«)ir 



Fig. 6. 



Survey No. 500. 
Mineral District No. 



3. 



PLAT of tlie claim of John H. Marshall upon the Excelsior Lode, 

California Mining District, Lake, County, State 

OF Colorado. Containing an area of. . . .5.74. . . .acres. Scale of , . . . 
200... feet to the inch. Variation. .. .East. Surveyed by Geo. D. 
Williams, U. S. Deputy Mineral Surveyor. 

October 28, 1881. 



The Original Field Notes of the Survey of the Claim of. 



upon the from which this plat has been made, have been examined 

and approved, aud are on file in this office ; and I hereby certify that they furnish such an 
accurate description of said Mining Claim as will, if incorporated into a patent, serve fully 
to identify the premises, and that such reference is made therein to natural objects and per- 
manent monuments as will perpetuate and fix the locus thereof. I further certify that the 
value of the labor and improvements placed thereon by the applicant, or his grantors, is not 
less than Five Hundred Dollars, and that said improvements consist of : 



as appears by the report of the Deputy Surveyor. And I further certify that this is a correct 
Plat of said Mining Claim, made in conformity with said field notes of the survey thereof. 

U. S. Surveyor General's Office, 

Denver, Colorado. 
18... 



XJ. S. Surveyor General for Colorado. 



A TABLE 

LOGARITHMS OF NUMBERS. 



Remark. ^In the following table, in the nine right hand 
columns of each page, where the first or leading figures change 
from 9's to O's, points or dots are introduced instead of the 
O's, to catch the eye, and to indicate that from thence the 
two figures of the Logarithm to be taken from the second 
column, stand in the next line below. 



2 



A TABLK OF LOGARITHMS FROM 1 TO 1U,0U0. 



N. 


1 I 


a 


3 


4 


5 


6 


7 


8 i 9 


D. 


loo oooooo 0434 


0868 


i3oi 


1734 


2166 


2598 


3029 


3461 


3891 


43s 


loi I 43a I 


4751 


5i8i 


5609 


6o38 


6466 


6894 


7321 


7748 


8174 


428 


loa : 8600 


9026 
3259 


Q45i 


9876 


•3oo 


•724 


1147 


1570 


1993 


24i5 


424 


io3 


012837 


368o 


4100 


4521 


4940 


536o 


5779 


6197 


6616 


419 


104 


7033 745 1 


7868 


8284 


8700 
2841 


9116 

3252 


953j 
3664 


9947 


•36i 


•775 


416 


io5 


021 189 


i6o3 


20X6 


2428 


4075 


4486 4896 


412 


106 


53o6 


5715 


6125 


6533 


6942 


735o 


77^7 


8164 


8571 


8978 


408 


IS 


o384 
033424 


0789 


•195 


•600 


1004 


1408 


1812 


2216 


2619 


3021 


404 


3826 


4227 


4628 


5029 


543o 


5830 


623o 


6629 


7028 


400 


109 


7426 


7825 


8223 


8620 


9017 


9414 


9811 


•207 


•602 


•998 


396 


no 1041 393 


1787 


5182 


2576 


2969 


3362 


3755 


4148 


4540 


4932 


It 


III 


5323 


5714 


6io5 


6495 
•38o 


6885 


7275 
1153 


7664 


8o53 


8442 


883o 


iia 


9218 


9606 
3463 


^i 


•766 


1538 


1924 


2309 


2694 


386 


ii3 053078 


423o 


4613 


^, 


5378 
9185 


5760 


6142 


6524 


382 


114 ! 6905 


7286 


7666 


8046 


8426 


9563 
3333 


9942 
3709 


•320 


379 


ii5 


060698 
4458 


1075 


1452 


1829 


2206 


2582 


2958 


4o83 


376 


116 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443 


7815 


372 


\\l 


8186 


8557 


8928 


9208 


9668 
3352 


••38 


•407 
4o85 


•776 


1145 


i5i4 


369 


071882 


225o 


2617 
6276 


2985 


3718 


445i 


4816 


5x82 


366 


119 


5547 


5912 


6640 


7004 


7368 


7731 


8094 


8457 


8819 


363 


1 20 


079181 


q543 
3i44 


35o3 


•266 


•626 


•987 


i347 


1707 


2067 


2426 


36o 


lai 


082785 


3861 


4219 


4576 


4934 


5291 


5647 


6004 


357 


132 


636o 


6716 


7071 


7426 


7781 


8i36 


8490 


8845 


9198 


9552 
3071 


355 


123 


0905 
093422 


•258 


•611 


•963 


i3i5 


1667 


2018 


2370 


2721 


35i 


124 


3772 


4122 


4471 


4820 


5169 


55i8 


5866 


62i5 


6562 


349 


125 


6010 
100^71 


7257 


7604 


7951 


8298 


8644 


8990 
2434 


9335 


9681 
3119 


••26 


346 


126 


0715 


1059 


i4o3 


1747 


2091 


2777 


3462 


343 


\ll 


3804 


4146 


4487 


4828 


5169 
8565 


55io 


585i 


6191 


6531 


6871 


340 


7210 


7549 


7888 


8227 


8903 


9241 


9579 


9916 


•253 


338 


129 


110590 


0926 


1263 


1599 


1934 


2270 


2605 


2940 


S275 


3609 


335 


i3o 


Ii3g43 


4277 


4611 


4944 


5278 


56ii 


5943 


6276 


6608 


6940 


333 


i3i 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


3198 


•245 


33o' 


l32 


1 205-14 


0903 


123l 


i56o 


1888 


2216 


2544 


2871 


3525 


328 


i33 


3832 


4178 


45o4 


483o 


5i56 


5481 


58o6 


6i3i 


6456 


6781 


325' 


1 34 


7io5 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 


••12 


323 


i35 


i3o334 


o65d 


0977 


1298 


1619 


If^t 


2260 


258o 


2900 


3219 
640J 


321 


1 36 


3539 


3858 


4177 


4496 


4814 


545 1 


5769 


6086 


3i8 


»37 
i38 


6721 


7037 


7354 


■7671 


7987 


83o3 


8618 


8934 


9249 


9564 


3i5 


0870 


•194 


•5o8 


•822 


ii36 


i45o 


1763 


2076 


2389 


2702 


3i4 


139 


14301 5 


3327 


3639 


3951 


4263 


4574 


4885 


5196 


55o7 


58i8 


3x1 


140 


146128 


6438 


6748 


7o58 


7367 


7676 


7985 


8294 


86o3 


8911 


309 


141 


9210 


9527 


9835 


•142 


•449 


•756 


io63 


1370 


1676 


1982 


307 


142 


152288 


2694 


2900 


32o5 


35io 


38i5 


4120 


4424 


4728 


5o32 


3o5 


143 


5336 


5640 


5943 


6246 


6549 


6852 


7i54 


7457 


7759' 


8061 


3o3 


144 


8362 


8664 


8965 


9266 


9567 


9868 


•168 


•469 


•769 


1068 


3oi 


145 


i6i368 


1667 


1967 


2266 


2564 


2863 


3i6i 


3460 


3758 


4o55 


299 


146 


4353 


465o 


4947 


5244 


5541 


5838 


6i34 


6430 


6726 


7022 


297 
295 


"^2 


7317 


7613 


7008 


8203 


8497 
1434 


8792 


9086 


9380 


9674 


9968 
2895 


170262 


o555 


0848 


1141 


1726 


2019 


23 1 1 


26o3 


293 


149 


3i86 


3478 


3769 


4060 


435 1 


4641 


4932 


5222 


55i2 . 5802 


291 


i5o 


I 7609 I 


638i 


6670 


6959 


7248 


7536 


7825 


8ii3 


8401 8689 


289 


i5i 


I8I844 


9264 


9552 


9839 


•126 


•41 3 


!^99 


•985 


1272 l558 


287 
285 


l52 


2129 

4975 


24i5 


2700 
5542 


2985 


3270 


35DD 


3839 


4123 4407 


1 53 


4691 


5259 


5825 


6108 


6391 


6674 


6956 7239 


283 


1 54 


7521 7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 ••5i 


281 


i55 


190332 0612 


0892 
368i 


1171 


i45i 


1730 
45i4 


2010 


2289 


2567 j 2846 


270 


i56 


3 1 25 3403 


3959 


4237 


4792 


5069 


5346 


5623 


278 


\ll 


5899 6176 


6453 


6729 


7oo5 


7281 


7556 


7832 


^IT 


8382 


276 


8657 8932 


9206 


9481 


9755 


••29 


•3o3 


•577 
33o5 


•85o 


1124 


274 


159 


201397 1670 


1943 


3216 


2488 


2761 


3o33 


3577 


3848 


272 


N. 





I 


2 


' 


4 


5 


6 


7 


8 


9 


D. 



TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 





I 


3 


3 


4 


5 


6 


7 


8 


9 


D. 


i6o 


204120 


4391 


4663 


4934 


5304 


5475 


5746 


6oi6 


6386 


6556 


271 


i6i 


6826 


7006 
9783 


7365 


7634 


7904 


Hl^ 


8441 


^21° 


X 


9247 


269 


162 


95i5 


••5 1 


•3i9 


•D86 


•853 


1121 


1888 


1931 

4679 


267 

266 


i63 


112188 


3454 


3730 

5373 


3986 


3252 


35i8 


3783 


4049 


43i4 


164 


4844 


5109 


5638 


5902 


6166 


643o 


6694 


6957 


7221 


264 


i65 


7484 


7747 


8010 


8273 


8536 


8798 


9060 


9823 


9585 


9846 ' 262 


166 


J10108 


0370 


o63i 


0893 


ii53 


1414 


1675 


1986 


3196 


2456 261 


;s 


3716 


2976 


3336 


3496 


3755 


401 5 


4374 


4588 


4792 


5o5i , 259 


5309 


5568 


5836 


6084 


6342 


6600 


6858 


7ii5 


7873 


7680 


258 


169 


7887 


8144 


8400 


8657 


8913 


9170 


9426 


9682 


9988 


•193 


256 


170 


230449 


0704 


0960 
35o4 


I3l5 


1470 


1724 


1979 
4517 


2284 


3488 


3743 


354 


171 


3996 
5528 


325o 


3757 
6285 


401 1 


4264 


4770 


5o33 


5376 


253 


172 


5781 


6o33 


6537 


6789 


7041 


7292 


7544 


7795 


353 


173 


8046 


8297 


8548 


8799 


9049 


9299 


9550 


9800 


••5o 


•3oo 


35o 


174 


240549 


0709 


1048 


I3Q7 


i546 


1795 


2044 


2298 


3541 


3790 


240 


175 


3o38 


3286 


3534 


37§3 


4o3o 


6745 


4525 


4773 


5019 


5366 


348 


176 


55i3 


5759 


6006 


6353 


6499 
8q54 
1395 


6991 


7287 


7482 


7728 


346 


172 

178 


7973 


8219 


8464 


8709 


9198 
i63» 


9443 


9687 

2125 


9982 
2868 


•176 


345 


25o420 


0664 


0008 


ii5i 


1881 


2610 


243 


179 


2853 


3096 


3338 


358o 


3822 


4064 


43o6 


4548 


4790 


5o3i 


243 


180 


255273 


55i4 


5755 


5996 
83o8 


6237 


6477 


6718 


6958 


7198 


7439 
9888 


341 


181 


7679 


7018 
o3io 


8i58 


8637 

1023 


8877 


9116 


9355 


9594 


389 


182 


260071 


0548 


0787 


1263 


i5oi 


1789 


1076 
4J46 


2214 


338 


1 83 


245i 


3688 


3935 


3i63 


3399 


3636 


3873 


4109 


4582 


337 


184 


4818 


5o54 


5390 


5525 


5761 


5996 


6232 


6467 


6702 


6987 


335 


i85 


7172 


7406 


7641 


7875 


811O 


8344 


8578 


8812 


9046 


9279 


384 


186 


95i3 


9746 


9980 


•3l3 


•446 


•679 


•912 


1144 


1877 


1609 


388 


188 


271842 


3074 
4389 


33o6 


3538 


2770 

5o8i 


3ooi 


3233 


8464 


8696 


8927 


383 


4i58 


4620 


485o 


53ii 


5542 


5772 


6002 


6282 


33o 


189 


6462 


6692 


6921 


7i5i 


7880 


7609 


7888 


8067 


8296 


8525 


339 


190 


278754 
281033 


8982 


9211 


943o 
1715 


9667 


9895 


•128 


•85i 


•578 


•806 


328 


191 


1261 


1488 


1942 


2169 


2896 
4656 


2623 


3849 


8075 


327 


192 


33oi 


3537 


3753 


3979 


42o5 


443 1 


4883 


5i07 


5832 


226 


193 


5557 


5783 


6007 


6232 


6456 


6681 


6905 


7180 


7354 


7578 


225 


194 


7802 


8036 


8249 


8473 


8696 


8920 


9143 


9866 


i8i3 


9812 


223 


195 


290035 


0357 


0480 


0703 


0925 


1 147 


1869 


1591 


2084 


222 


196 


2256 


3478 
4687 


2699 


2920 


3i4i 


3363 


3584 


38o4 


4025 


4246 


221 


197 


4466 


4907 


5i27 


5347 


5567 


5787 


6007 
8.9^ 


6226 


6446 


220 


198 


6665 


6884 


7104 


7333 


7542 


7761 


7979 


8416 


8635 


210 


199 8853 


9071 


9289 


9507 


9725 


9943 


•161 


•378 


•595 


•8i3 


218 


200 


3oiu3o 


1347 


1464 


1681 


1898 


2114 


233 1 


3547 


3764 


2980 


317 


201 


3196 
535 1 


3413 


3628 


3844 


4069 


4275 


4491 
6689 


4706 


4931 


5i36 


3l6 


202 


5566 


5781 


5996 


62H 


6425 


6854 


7068 


7282 


3l5 


203 


9630 


7710 

9843 


7924 


8137 


835i 


8564 


8778 


8991 


9304 


9417 


3l3 


204 


••56 


•368 


•481 


•693 


•906 


1118 


i33o 


1 542 


3131 


2o5 


311754 


1966 


2177 

4389 


3389 


2600 


2812 


8028 


3384 


3445 


8656 


311 1 


206 


3867 


4078 
6180 


4499 


4710 


4920 


5i3o 


5340 


555i 


5760 


3I0{ 


a 


5970 


6390 


6599 


6809 
8898 


7018 


7227 


7436 


7646 


7854 


3091 

3o8, 


8o63 


8373 
o354 


8481 


8689 


9106 


9314 


9533 


9780 


9988 


ao9 


320146 


o563 


0769 


0977 


1 184 


1891 


1598 


i8o5 


2012 


207 


aio 


322219 


3436 


3633 


3839 


3046 


3252 


3458 


3665 


3871 


tru 


206 


211 


4282 


4488 


4694 


4899 
6930 


5io5 


53io 


55i6 


5731 


5936 


2o5 


312 


6336 


6541 


6745 


7155 


735n 


7563 


7767 
9805 


7973 


8176 


204 


ai3 


838o 


8583 


8787 


8991 


9194 


9898 


9601 


•••8 


•211 


2o3 


214 


33o4i4 


0617 


0819 


1033 


1225 


1427 


1680 


1883 


3084 


2236 


202 


2l5 


2438 


3640 


3843 


3o44 


3246 


3447 


8649 


385o 


4o5i 


4253 i 


202 


216 


4454 


4655 


4856 


5o57 


5257 


5458 


5658 


5859 


6059 


6260 


30I 


III 


6460 


6660 


6860 7060 


7260 


7459 


7659 


7858 


8o58 


8257 
•246 1 


200 


8456 


8656 


8855 


9054 


9253 


945 1 


9650 


9849 


••47 

2028 




ti9 


340444 


0643 


0841 


1039 


1237 


1435 


i632 


i83o 


2225 1 


N. 





• 


2 


3 

1 


4 




5 


6 


7 


8 


9 



16 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 


oil 2 


3 


4 


5 


6 


7 


8 


9 


D. 


220 


342423' 2620 J817 
4392 4589 , 4785 


3oi4 


3212 


3409 


36o6 


38o2 


3999 


4196 


196 


221 


4981 


5178 

7i35 


5374 


5570 


5766 


5962 


6i57 


222 


6353 6549 


6744 


6939 


7330 


7525 


7720 


7015 


8x10 


195 


223 


83o5 85oo 


8694 


8889 


9083 


9278 


9472 


9666 


9860 


••54 


194 


224 


350248 0442 


o636 


0829 


X023 


1216 


1410 


i6o3 


1796 


1989 


193 


225 


2i83 2375 1 2568 


2761 


2954 


3x47 


3339 


3532 


3724 


3916 


193 


226 


4io8j 43oi 1 4493 


4685 


4876 


5o68 ' 5260 


5452 


5643 


5834 


192 


227 
228 


6026 6217 
7935; 8125 


6408 


6599 


6790 


698X 7x72 


7363 


7554 


7744 


191 


83i6 


85o6 


8696 


8886 


9076 


9266 


9456 J 9646 


190 
189 


! 229 


9835; ••25 


•2l5 


•404 


•593 


•783 


•972 


ix6x 


i35o 


1 539 


23o 


361728 1917 


2io5 


2294 


2482 


267 X 


2859 


3o48 


3236 


3424 


188 


23 1 


36i2; 38oo 


3988 


4176 


4363 


455x 


4739 


4926 


5xx3 


53oi 


. 188 


232 


5488! 5675 


5862 


6049 


6236 


6423 


66x0 


6796 


6983 


7x69 


18^ 


233 


73561 7542 


7729 
9687 


7915 


8101 ' 8287 
9o58 '■ •i4i 


8473 


8659 
•5x3 


8845 


9o3o 


234 


92161 9401 


9772 
1622 


•328 


•698 


•883 


i85 


235 


371068 


1253 


1437 


j^o6 


1991 
383 X 


2175 


236o 


2544 


2728 
4565 


184 


236 


2912 


3096 
4932 


3280 


3464 


«5647 


401 5 


4198 


4382 


184 


^^1 
238 


4748 


5ii5 


5298 : 5481 


5664 


5846 


6029 


6212 


6394 


i83 


6577 
8398 


6759 
858o 


6942 


7124 
8943 


7306 


7488 


7670 
9487 


7852 


8o34 


8216 


182 


239 


8761 


9124 


9306 


9668 


9849 


••3o 


181 


240 


380211 


0392 


0573 


0754 
2557 
4353 


0934 


ixx5 


1296 


1476 


1 656 


1837 
3636 


181 


241 


2017 
38i5 


3ga5 


2377 


2737 


29x7 


3097 


3277 


3456 


180 


242 


4174 


4533 


4712 


4891 


5070 
6856 


5249 


5428 


\]t 


243 


56o6 


5785 
7568 


5964 


6142 


632 X 


6499 


6677 
8456 


7o34 
881 X 


7212 
8989 


244 


7390 


7746 
9520 


7923 


8xox 


8279 


8634 


178 


245 


9166 


9343 


9698 


9875 


••5 1 


•228 


•4o5 


•582 


•759 

2521 


176 


246 


390935 


1112 


1288 


1464 


1641 


18x7 
3575 


\n^ 


2169 


2345 


247 


2697 


2873 


3o48 


3224 


3400 


375x 
55oi 


3926 


410X 


6025 


176 


248 


4452 


4627 


4802 


4977 


5x52 


5326 


5676 


585o 


175 


249 


6199 


6374 


6548 


6722 


6896 


7071 


7245 


7419 


7592 


7766 


174 


25o 


397940 


8114 


8287 


8461 


8634 


8808 


898X 


9154 


9328 


95ox 


173 


25l 


9674 


9847 


••20 


•192 


•365 


•538 


•71X 


•883 


xo56 


X228 


173 


352 


401401 


1573 


1745 


1917 
3635 


2089 


226X 


2433 


26o5 


2777 


2949 
4663 


172 


253 


3l2I 


3292 


3464 


3807 


3978 
5688 


4149 


4320 


4492 


171 


254 


4834 


5oo5 


5176 


5346 55x7 


5858 


6029 


6x99 


6370 


171 


255 


6540 


6710 


6881 


705 1 
8749 


7221 


7391 


756x 


7731 


7001 
95o5 


8070 


109 


256 


8240 


8410 


8579 


89x8 


9087 


9207 


9426 


9764 


257 
258 


9933 


•102 


•271 


•440 


•60Q 
2293 


•777 


•946 


1114 


1283 


X45i 


169 


41 1620 


1788 


1956 


2124 


246X 


2629 


2796 


2964 


3i32 


160 


259 


33oo 


3467 


3635 


38o3 


3970 


4x37 


43o5 


4472 


4639 


4806 


167 


260 


414973 


5i4o 


5307 
6973 


5474 
8798 


5641 


58o8 


5974 
7638 


6141 


63o8 


6474 
8x35 


166 


261 


6641 


6807 


7306 


7472 


7804 


7970 


262 


83oi 


8467 


8633 


8964 


9129 


9295 


9460 


9625 


9701 
1439 


i65 


263 


9956 


•121 


•286 


•451 


«*6x6 


•781 


•945 


IIIO 


1275 


i65 


264 


431604 


1788 


1933 


2097 
3737 


226X 


2426 


2590 


2754 
4392 


2918 
4555 


3082 


164 


265 


3246 


3410 


3574 


3901 


4o65 


4228 


4718 


164 


266 


4882 


5o45 


5208 


5371 


5534 


5697 


586o 


6023 


6x86 


6340 
7973 
9591 


i63 


267 
268 


65ii 


6674 


6836 


6999 


7x61 
8783 


7324 
8944 


7486 


7648 


7811 


162 


8x35 


8297 


8459 
••75 


8621 


9106 


9268 


9429 


162 


»69 


9752 


9914 


•236 


•398 


•559 


•720 


•881 


1042 


X203 


161 


270 


43 I 364 


i525 


i685 


1846 


2007 


2167 


2328 


2488 


2649 


2809 


161 


271 


6i63 


3i3o 


3290 


3450 


36x0 


3770 


3930 


4090 
5685 


4249 


4409 


160 


272 


4729 


4888 


5048 


5207 
6798 
8384 


5367 


5526 


5844 


6004 


i59 


273 


6322 


6481 


6640 


6957 


7xx6 
870X 


7275 
8859 


7433 


7592 


i58 


274 


7751 
9333 


7909 


8067 8226 


8542 


9017 


9175 
•752 

2323 


275 


9491 


9648 9806 


9964 


•122 


•279 
i852 


•437 


•594 


1 58 


276 


440909 


1066 


1224 i38i 


i538 


1695 


2009 


2x66 


i57 


V,l 


2480 


2637 


2793 


2050 

45x3 


3 106 


3263 


3419 


3576 


3732 


3889 


[U 


4045 


4201 


4357 


4669 


4825 


4981 
6537 


5x37 


5293 5449 
6848 7oo3 


279 
N. 


56o4 


6760 


5915 


6071 


6226 


6382 


6692 


1 55 
D. 





I 

L 


2 


3 


4 


5 


6 


7 


8 9 



A TABLE OF LOGARITHMS FBOM 1 TO 10,000. 



N. 





I 


3 


3 


4 


5 


6 


7 


8 


9 


D. 


a8o 


447158 
8706 


73i3 


7468 


7623 


7778 


7933 


8088 


8242 


83^7 


8552 


1 55 


a8i 


8861 


9015 


9170 


9324 


9478 


9633 


9787 


9941 


••95 
1 633 


1 54 


383 


450249 


. o4o3 


0557 


071 1 


0865 


1018 


1172 


1826 


1479 


i54 


283 


1786 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


8012 


3i65 


1 53 


384 


33i8 


3471 


3624 


3777 


3930 


4082 


4235 


4887 


4540 ' 4693 


1 53 


385 


4845 


4997 


5i5o 


53o2 


5454 


56o6 


5758 


5910 


6062 1 6314 


1 53 


386 


6366 


65i8 


6670 


6821 


6973 


7125 


7276 
8789 


7428 


7579 


7781 


i5a 


387 ■ 7882 


8o33 


8184 


8336 


8487 


8638 


8940 


9091 


9242 


i5i 


38a 


9392 


9543 


9694 


9845 


9995 


•146 


•296 


•447 


•597 


•748 


i5i 


289 


460898 


1048 


1 198 


1348 


1499 


1649 


1799 


1948 


2098 


2248 


i5o 


390 


462398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 


i5o 


391 


3893 


4042 


4191 


4340 


4490 


4639 


4788 


4986 


5o85 


5234 


149 


292 


5383 


5532 


568o 


5829 


5977 


6126 


6274 


6428 


6571 


6719 


149 


293 


6868 


7016 


7164 


73i2 


7460 


7608 


7756 


7904 


8o52 


8200 


148 


294 


8347 


8495 


8643 


8790 


8938 


9085 


9233 


9880 


9527 


9675 


148 


295 


9822 


9969 


•116 


•263 


•410 


•557 

2020 


•704 


•85i 


•998 


1 145 


147 


296 


471292 


1438 


1 585 


1732 


1878 


2171 


2818 


2464 


2610 


146 


297 


2736 


2903 


3o4Q 


3195 


3341 


3487 


3633 


3779 


8925 
5881 


f 4071 


146 


298 


4216 


4362 


45o8 


4653 


4799 


4944 
6897 


5090 


5285 


5526 


146 


299 


5671 


58i6 


5962 


6107 


6202 


6542 


6687 


6882 


6976 


145 


3oo 


477121 
8566 


7266 

871 1 


741 1 
8855 


7555 
^999 


7700 


7844 


7989 


8188 


8278 


8433 


145 


3oi 


9143 


9287 


9431 


9575 


9719 


9868 


144 


3o2 


480007 


oi5i 


0294 


0438 


o582 


0725 


0869 


1012 


ii56 


1299 
2781 


144 


3o3 


1443 


1 586 


1729 


1872 


2016 


2159 


2802 


2445 


2588 


143 


3o4 


2874 


3oi6 


3i59 


33o2 


3445 


3587 


8780 


8872 


401 5 


4157 


143 


3o5 


43oo 


4442 


4585 


4727 


4869 


5oii 


5i53 


5295 


5437 


5579 


142 


3o6 


5721 


5863 


6oo5 


6147 


6289 


643o 


6572 


6714 


6855 


6997 


142 


3o7 


7i38 


7280 


7421 


7563 


7704 


7845 


7086 
9896 


8127 


8269 


8410 


141 


3o8 


855i 


8692 


8833 


8974 


9114 


9255 


9587 


9677 


9818 


141 


3o9 


9958 


••99 


•289 


•38o 


•520 


•661 


•801 


•941 


1081 


1233 


140 


3io 


491362 


l502 


1642 


1782 


1922 


2062 


2201 


2841 


2481 


3631 


140 


3ii 


2760 


2900 


3o4o 


3179 


3319 


3458 


8 597 


3787 


3876 


401 5 


189 


3l2 


4i55 


4294 


4433 


4572 


471 1 


485o 


4989 


5x28 


5267 


5406 


189 


3i3 


5544 


5683 


5822 


5960 


6099 


6238 


6876 


65i5 


6653 


6791 


189 


3i4 


6930 
83ii 


7068 


7206 


7344 


7483 


7621 


77^9 


7897 


8o35 


8173 


i38 


3i5 


8448 


8586 


8724 


8862 


8999 
•374 


9187 


9275 


9412 


9550 


1 38 


3i6 


9687 


9824 


9962 


••99 


•236 


•5ii 


•648 


•785 


•922 


187 


3i7 


5oio59 


1196 


1333 


1470 


1607 


1744 


1880 


2017 


2i54 


2291 


187 


3i8 


2427 


2564 


27CO 


2837 


2073 
4335 


3109 


8246 


3382 


35i8 


3655 


i36 


319 


3791 


3927 


4o63 


4199 


4471 


4607 


4743 


4878 


5oi4 


186 


330 


5o5i5o 


5286 


5421 


5557 


5693 


5828 


5964 
7816 


6099 


6234 


6870 


1 36 


331 


65o5 


6640 


6776 


691 1 


7046 


7181 


7451 


7586 
8984 


7721 


135 


332 


7856 


1% 


8126 


8260 


8395 


853o 


8664 


8799 


9068 


i35 


323 


9203 


9471 


9606 


9740 


9874 


•••9 


•143 


•277 


•4n 


i34 


324 


5io545 


0679 


o8i3 


0947 


I081 ' I2l5 


1849 


1482 


1616 


1750 


i34 


325 


1 883 


2017 


2l5l 


2284 


2418 ! 255l 


2684 


2818 


2951 


8084 


;33 


326 


3218 


3351 


3484 


3617 


3750 3883 


4016 


4149 


4382 


4414 


1 33 


337 


4548 


4681 


48i3 


4946 


5079 ! 52 1 I 

64o3 6535 


5844 


5476 


5609 


5741 


i33 


328 


5874 


6006 


6139 


6271 


6668 


6800 


6983 


7064 


182 


339 


7196 


7328 


7460 


7592 


7724 


7855 


7987 


8119 


835i 


8382 


182 


33o 


5i85i4 


8646 


8777 8909 


9040 


9171 


9808 


9434 


9566 


9697 


i3i 


33i 


9828 


9959 


••90 ' •221 


•353 


•484 


•6i5 


•745 


•876 


1007 


i3i 


332 


521138 


1269 
2575 


1400 1 i53o 


1661 


1792 


1922 


2o53 


3i83 


2814 


i3i 


333 


2444 


2705 2835 


2966 


3096 


8226 


8856 


3486 


36i6 


i3o 


334 


3746 


3876 


4006 , 4i36 


4266 


4396 


4526 


4656 


4785 


4915 


i3o 


335 


5o45 


5174 
6409 


53o4 ■ 5434 


5563 


5693 


5822 


5951 


6081 1 6210 j 


129 


336 


6339 


6598 , 6727 
7888 1 8016 


6856 


6985 


7II4 


7243 


7872 


7501 


129 


337 

m 


7630 


7759 
9045 


8145 


8274 


8403 


853 1 


8660 


8788 


128 


9174 1 


9302 


9430 


9559 


9687 


9815 


9943 


••72 


339 


53o2oo 


o338 


0456 i 


o584 


0713 ; 0840 I 

i 


0968 


109/* 

"7 1 


1228 


i85i 


138 


N. 





1 


3 


3 


4 


5 


6 


8 

-1 


9 



a 



A. TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 
340 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


531479 
2754 


1607 


1734 


1862 


i'990 


2117 


2345 


2372 


25oo 


2627 


128 


341 


2882 


3009 


3i36 


3264 


3391 


35i8 


3645 


3772 


3899 


127 


342 


4026 


41 53 


4280 


4407 


4534 


4661 


4787 


4914 


5o4i 


5167 


"2 
126 


I 343 


5204 


5421 


5547 


5674 
6937 


58oo 


5927 


6o53 


6180 


63o6 


6432 


! 344 


6558' 6685 


681 1 


7063 


V^2 
8448 


73i5 
8574 


7441 
8699 
9954 


^82? 


J^?I 


126 


! 345 


7819 


7945 


8071 


8197 
9452 


8322 


126 


346 


9076 


9202 


9327 


9578 


9703 


9829 


••79 


•204 


125 


¥2 


* io329 


0455 


o58o 


0705 


o83o 


0955 


1080 


I205 


i38o 


1454 


125 


348 


'in 
1825 


1704 


1829 


1953 


2078 


2203 


2827 


2452 


2576 


2701 


125 


349 


2950 


3074 


3199 


3323 


3447 


3571 


8696 


3820 


J944 


124 


350 


144068 


4192 


43i6 


4440 


4564 


4688 


4812 


4986 


5o6o 


5i83 


124 


35i 


5307 


543 1 


5555 


5678 


58o2 


5925 


6049 


6172 


6296 


6419 


124 


352 


6543 


6666 


6789 


6913 


7086 


7159 
8389 


7282 


7405 

8685 


7529 
875s 


7652 
8881 


123 


353 


7775 


7898 


8021 


8144 


8267 


85i2 


123 


354 


9003 


9126 


9249 


9371 


9494 


9616 


9789 


9861 


9984 


•106 


123 


355 


550228 


o35i 


0473 


0595 


0717 


0840 


0962 


1084 


1206 


1828 


122 


356 


i45o 


1572 


1694 


1816 


1938 


2060 


2181 


2808 


2425 


2547 


122 


357 
358 


2668 


2790 


291 1 


3o33 


3i55 


3276 


3398 


35i9 


8640 


8762 


121 


3883 


>loo4 


4126 


4247 


4368 


4489 


4610 


4781 


4852 


iul 


121 


359 


5094 


52i5 


5336 


5457 


5578 


5699 


5820 


5940 


6061 


121 


360 


5563o3 


6423 


6544 


6664 


6785 


6905 


7026 
8228 


7146 
8840 


7267 


7337 


120 


36i 


7507 
8709 


8829 


7748 
8948 


7868 


7988 


8108 


8469 


8589 


120 


362 


9068 


9188 


9808 


9428 


9548 


9667 
•863 


9787 


120 


563 


9907 


••26 


•146 


•265 


•385 


•5o4 


•624 


•748 


•982 


119 


364 


56IIOI 


1221 


1 340 


1459 


1578 


1698 


1817 


1986 


2o55 


2174 


119 


365 


2293 


2412 


253i 


265o 


2769 


2887 


3oo6 


3i25 


3244 


8362 


119 


366 


3481 


36oo 


3718 


3837 


3955 


4074 


4192 


48ii 


4429 


4548 


119 


367 
368 


4666 


4784 


4903 


502I 


5i39 


5257 


5376 
6555 


5494 


56i2 


5780 


110 


5848 


5966 


6084 


6202 


6320 


6437 


6678 


6791 


6909 


118 


369 


7026 


7144 


7262 


7379 


7497 


7614 


7732 


7849 


7967 


8084 


118 


370 


568202 


83i9 


8436 


8554 


8671 


8788 


8905 


9028 


9140 


9257 


117 


371 


9374 


9491 


9608 


9725 


9842 


9959 


••76 


•198 


•809 


•426 


117 


372 


570543 


0660 


0776 


0893 


lOIO 


1 1 26 


1243 


1359 


1476 


i5q2 


117 


373 


1709 


1825 


1942 


2o58 


2174 


2291 


2407 
3568 


2523 


2689 


2765 


116 


374 


2872 


2988 


3io4 


3220 


3336 


3452 


8684 


3800 


3915 


116 


375 


4o3i 


4147 


4263 


4379 


4494 


4610 


4726 


4841 


4957 


5072 


116 


376 


5i88 


53o3 


5419 


5534 


5650 


5765 


588o 


5996 


6111 


6226 


ii5 


m 


6341 
7492 


6457 
7607 


6572 
7722 


6687 
7836 
8983 


6802 
7951 


8?66 


7082 
8181 


7147 
8295 


7262 
8410 


nil 


ii5 
ii5 


379 


8639 


8754 


8868 


9097 


9212 


9826 


9441 


9555 


9669 


114 


380 


579784 
580925 


9898 
10J9 


••12 


•126 


•241 


•355 


•469 
1608 


•583 


•697 
.836 


•811 


114 


38i 


ii53 


1267 


i38i 


i4q5 
263 1 


1722 
28.58 


1950 


114 


382 


2o63 


2177 


2291 


2404 


25i8 


2745 


'-972 


3o85 


114 


383 


433 1 


33i2 


3426 


3539 


3652 


3765 
4896 


3879 


3992 


4io5 


4218 


ii3 


384 


WiA 


4557 


4670 


4783 


5009 


5l22 


5285 


5348 


ii3 


385 


5461 


5574 


5686 


5799 
6925 


5912 


6024 


6187 


6250 


6862 


6475 


ii3 


386 


6587 


6700 


68i2 


7037 


7U9 
8272 


7262 
8384 


8496 


7486 
8608 


7599 
0720 
9888 


112 


387 
388 


7711 
8832 


7823 
8944 


7935 


8047 


8160 


112 


9o56 


9167 


9279 


9391 


95o3 


9615 


9726 
•842 


112 


389 


995d 


••bi 


•173 


•284 


•396 


•5o7 


•619 


•780 


•953 


iia 


390 


591065 


1176 

2288 


1287 


1399 


i5io 


1621 


1782 

2843 


1848 


1^55 


2066 


Hi 


391 


2177 


2399 


25lO 


2621 


2732 


2954 


8064 


8175 


III 


392 


3286 


3397 


35o8 


36i8 


3729 


3840 


3950 


4o6i 


4171 


4282 


III 


393 


4393 


45o3 


4614 


4724 


4834 


4945 


5o55 


5i65 


5276 


5886 


110 


394 


5496 


56o6 


5717 


5827 


5987 


6047 


6157 


6267 


6377 


6487 


HO 


395 


6597 
2695 

9883 


6707 


6817 


6927 


7037 


7146 
8243 


7256 


7866 


7476 
8572 


7586 
8681 


110 


396 


7805 
8900 


7914 


8024 


8i34 


8858 


8462 


no 


in 


9009 


91 19 


9228 


9337 


9446 


9556 


9665 


X 


109 


398 


9992 


•101 


•210 


•319 


•428 


•537 
1625 


•646 


•755 
1843 


109 


399 


600973 


1082 


1191 
2 


1299 


1408 


i5i7 


1734 


I95I 


109 


N. 


3 


I 


3 


4 


5 


6 


7 


8 


9 


D. 



A TABLE OF LOGARITHMS FROM ] TO 10,000. 



N. 



602060 
J144 
4226 
53o5 



400 
401 
402 
4o3 

404 , 

405 1 7455 

406 d526 



2169 
3253 
4334 
54i3 



409 

410 
411 
412 

4i3 
414 
41 d 
416 

41 B 
419 
420 
421 
422 
423 
424 
425 
426 

427 
428 
429 

43o 
43 1 
432 
433 
434 
435 
436 
437 
438 
439 
440 
441 
442 
443 
444 
445 
446 
447 
448 

449 
45o 
45 1 
452 
453 
454 
456 
456 



538i 6489 
7562 



459 
N. 



9594 

610660 

1723 

612784 
3842 
4897 
5960 
7000 
8048 
9093 

620136 
1176 
2214 

623249 
4282 
53i2 
6340 
7366 
8389 
9410 

630428 
1444 
2457 

633468 

4477 
5484 
6488 
7490 
8489 
94S6 
640481 
1474 
2465 

643453 
4439 
5422 
6404 
7383 
836o 
9335 

65o3o8 
1278 
2246 

653213 

5i38 
6098 
7056 
801 1 
8965 

660865 
i8i3 



8633 
9701 
0767 
1829 

2890 



5ooc 
6o55 
7io5 
8i53 
9198 
0240 
1280 
23i8 

3353 
4385 
541 5 
6443 
7468 

8491 
95i2 
o53o 
1545 
2559 

3569 
4578 
5584 
6588 
7590 
8589 
9586 
o58i 
1573 
2563 

355i 
4537 
5521 
65o2 
7481 
8458 
9432 
o4o5 
1375 
2343 

33oo 
4273 
5235 
6194 
7152 
8107 
9060 
••11 
0960 
1907 



2277 
336i 
4442 
5521 
6596 
7669 
8740 
9808 
0873 
1936 



5io8 
6160 
7210 
S257 
9302 
0344 
i384 
2421 

3456 
4488 
55i8 
6546 
7571 
8593 
9613 
o63i 
1647 
2660 

3670 
4679 
5685 
6688 
7690 
8689 
9686 
0680 
1672 
2662 

365o 
4636 
5619 
6600 

8555 
953o 
o5o2 
1472 
2440 

34o5 
4369 
533 1 
6290 

7247 
8202 
9155 
•106 
io55 
2002 



2386 
3469 
455o 
5628 
6704 

IIV^ 

9914 
0979 
2042 

3l02 

4159 
52i3 
6265 



9406 
0448 
1488 

2525 

3559 
4591 
5621 
6648 

7673 
8695 
97i5 
0733 
1748 
2761 

3771 

5785 
6789 
7700 
8780 

9785 
0779 
1771 
2761 

3749 
4734 
5717 
6698 
7676 
8653 
9627 
0599 
1569 
2536 

35o2 
4465 
5427 
6386 
7343 
8208 
9260 
•201 
ii5o 
2096 



1494 

4658 
5736 
681 1 
7884 
8954 
••21 
1086 
2148 

3207 
4264 
53i9 
6370 



73 1 5 7420 
8362 I 8466 



9511 
o552 
1592 
2628 

3663 
4695 
5724 
6751 

7775 
8797 
9817 
o835 

1849 
2862 

3872 
4880 
5886 



7890 
8888 
9885 
0879 
1871 
2860 

3847 
4832 
58i5 
6796 
7774 
8750 
9724 
0696 
1666 
2633 

3598 
4562 
5523 
6482 
7438 
8393 
9346 
•296 
1245 
219] 

4 



s6o3 
3686 
4766 
5844 
6919 

i 7991 
9061 
•128 

I 2254 

33i3 
4370 
5424 
I 6476 
7525 
8571. 
9615 
o656 
i6q5 
2732 

3766 
4798 
5827 
6853 
7878 
8900 
9919 
0986 
1951 
2963 

3973 
4981 
5986 
6989 

7990 
8988 

9984 
0978 
1970 
2959 

3946 
4931 
5913 
6894 
7872 
8848 
9821 
0793 
1762 
2730 

3695 
4658 
5619 
6577 
7534 
8488 
9441 
.301 
1339 
2286 



2711 
3794 
4874 
5961 
7026 
8098 
9167 
•234 
1298 
236o 

3419 

447^ 



9719 
0760 
1799 
2835 

3869 
4901 
5929 
6956 
7980 
9002 
••21 
io38 

2052 
3064 

4074 

5o8i 
6087 
7089 
8090 
9088 
••84 
1077 
2069 
3o58 

4044 
5029 
601 1 
6992 
7969 
8945 
9019 
0890 
1859 
2826 

3791 
4754 
5715 
6673 
7629 
8584 
9536 
•486 
1434 
238o 



2819 
3902 
4982 
6o5o 
7133 
8205 
9274 
•341 
i4o5 
2466 

3525 
458i 



5529 I 5634 
658i I 6686 
7620 i 7734 
8676 8780 



9824 
0864 
1903 
2939 

3973 
5004 
6o32 
7o58 
8082 
9104 
•123 
1189 
2i53 
3i65 

4175 
5182 
6187 
7189 
8190 
9188 
•i83 
1177 
2168 
3i56 

4143 
5127 
6110 
7089 
8067 
9043 
••16 
0987 
1956 
2923 

3888 
485o 
58io 
6769 
7725 
8679 
9631 
•58i 
1 529 
2475 



2928 
4010 
5089 
6166 
7241 
83i2 
938i 

•447 
i5ii 
2572 

363o 
4686 
5740 

% 

8884 
9928 
0968 
2007 
3o42 

4076 
5107 
6i35 
7161 
8i85 
9206 
•224 
1241 

2255 

3266 

4276 
5283 
6287 

7290 

8200 
9287 
•283 

1276 
2267 
3255 

4242 
5226 
6208 

8i65 
9140 
•ii3 
1084 
2o53 
3oi9 

3984 
4946 
5906 
6864 
7820 
8774 
9726 
•676 
1623 
2569 

8 



3736 

4792 
5845 
6895 
7943 
.8989 

••32 

1072 
2110 
3146 

4179 

5210 

6238 
7263 
8287 
9308 
•326 
1 342 
2356 
3367 

4376 
5383 
6388 
7390 
8389 
9387 
•382 
1375 
2366 
3354 

4340 
5324 
63o6 
7285 
8262 
9237 
•210 
1181 

2l5o I 

3ii6 I 

408c i 

5o42 i 

6002 I 

6960 i 

7916 

8870 

9821 

•771 

1718 

2663 



D. 

:.o8 
lod 
108 
108 
107 
107 
107 



3o36 

4118 

5197 

6274 

7348 

8419 

9488 

•554 i 107 

1617 j 106 

2678 ; 106 

106 
106 
io5 
io5 
io5 
io5 
104 
104 
104 
104 

io3 
io3 
io3 
io3 
102 
102 
102 
102 

lOi 

101 



100 

100 

100 

100 

99 

99 

99 

99 

99 

99 

9^ 

9? 

95 
96 

97 
97 
97 
97 
97 
96 

95 
^? 

95 
D. 



8 



A TABLE OF LOGARITHMS FBOM 1 TO 10,000. 



N. 


° 


I 


3 


3 


4 


5 


6 


7 


8 


9 


D. 


460 


662758 


2852 


2047 


3o4i 


3i35 


323o 


3324 


3418 


35i2 


3607 


94 


461 


3701 


3795 

4735 


3S89 


3983 


4078 


4172 


4266 


436o 


4454 4548 


94 


46 a 


4642 


483o 


4924 
5862 


5oi8 


5lI2 


5206 


IIV, 


5393 : 5487 


94 


463 


558i 


5675 


5769 


5956 


6o5o 


6143 


6331 ! 6424 


94 


464 


65i8 


6612 


6700 


6799 


6892 


6986 


^1? 


7173 


7266 ] 7360 
8199 1 8293 


94 


465 


7453 


7546 


7640 
8572 


8665 


7826 


792c 
8852 


8106 


93 


466 


83d& 


8479 


8759 


8945 
9875 


9o38 


9i3i 9224 


93 


46t 


931-' 


9410 


95c3 


9596 


9689 


97S2 




••60 •'53 


93 


46d 


670546 0339 


043 1 


o524 


0617 


0710 


0802 


0695 


0988 i ic8o 


93 


469 


1173 1265 


i358 


i45i 


1 543 


i636 


1728 


1821 


1913 1 3005 


93 


470 


^72098 2190 


2283 


2375 


2467 


25t>0 


2652 


2744 


3836 2929 


95 


4-1 


302I 


3ii3 


32o5 


3297 


3390 


3482 


3574 


3666 


3758 3b5o 


92 


472 


3o42 
4861 


4o34 


4126 


4218 


43io 


4402 


4494 


4586 


4677 


4769 


93 


4-3 


4953 


5o45 


5i37 


5228 


5320 


5412 


55o3 


5595 


5687 


92 


474 


m 


5370 


5962 
6876 
7789 


6o53 


6145 


6236 


6328 


6419 


65ii 


6603 


92 


475 


6785 


6068 


7059 


7i5i 


7242 


7333 
8245 


7424 


7516 
8427 


91 


476 


7607 


nbgS 
8609 


7881 


7972 
8882 


8o63 


81 54 


8336 


9> 


477 


85i8 


8700 


8791 


8973 


9064 


9155 


9246 


9337 


9* 


47^ 


9428 


9519 


9610 


9700 


9791 


9882 


9973 


••63 


•i54 


•243 


9» 


479 


68o336 


0426 


o5i7 


0607 


0698 


0789 


0879 


0970 


1060 


ii5i 


91 


43o 


681241 


i332 


1432 


i5i3 


i6o3 


1693 


1784 


1874 


1064 


2o55 


90 


4S1 


2145 


2235 


2326 


2416 


25o6 


2596 


26S6 


2777 


2867 


3057 


90 


482 


3 047 


3 137 


3227 


3317 


3407 


3497 


3587 


3677 


3767 


3857 


90 


483 


3q47 
4845 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


90 


484 


4935 


5o25 


5ii4 


5204 


5294 
6189 


5383 


5473 


5563 


5652 


% 


485 


5742 


5-3i 


5921 
68i5 


6010 


6100 


6279 


6368 


6458 


6547 


486 


6636 


6726 


6904 


ti 


7o33 


7172 
8064 


7261 
8i53 


735i 


7440 
833i 


89 


48? 


7529 
8420 


7618 


8598 
9486 


7796 


^5 


8242 


89 


85o9 


86^7 


8776 


8o53 


9042 


9i3i 


9220 


89 


489 


9309 


9398 


9575 


9664 


9753 


9841 


9930 


••19 


•107 


89 


490 


690196 
1081 


0285 


0373 


0462 


o55o 


0639 


0728 


0816 


0905 


0093 


80 


491 


1170 


1258 


i347 


1435 


i524 


1612 


1700 


1789 


1877 


88 


492 


1965 

2847 


2o53 


2142 


223o 


23i8 


2406 


2494 


2583 


2671 


2759 


88 


493 


2935 
38i5 


3o23 


3iii 


3199 

4078 


3287 


3375 


3463 


355i 


3639 


88 


494 


3727 


3903 


3991 


4166 


4254 


4342 


443o 


4517 


88 


495 


46o5 


4693 


4781 


4S63 


4956 


5o44 


5i3i 


5219 


5307 


53q4 


88 


496 


5482 


556q 


5657 


5744 


5832 


5919 


6007 


6094 


6182 


6369 


87 


497 


6356 


6444 


653 1 


6618 


6706 


6793 


6880 


W9 


7o55 


r. 


?" 


498 


7229 


7317 
S188 


7404 


7491 


7578 


7663 


7752 


7926 
8796 


87 


499 


8101 


8275 


8362 


8449 


8535 


8622 


8709 


8883 


87 


5oo 


"^11 


9057 


9144 


923i 


9317 


9404 


9491 


9578 


9664 


975i 


87 


5oi 


9924 


••11 


••93 


•184 


•271 


•3d8 


•444 


•53 1 


•617 


87 


502 


700704 


0790 


0877 


0963 

1827 


io5o 


Ii36 


1232 


i3o9 


1395 


1483 


86 


5o3 


iks 


1604 


1741 


1913 


;^ 


2086 


2172 


2238 


3344 


86 


5o4 


243 1 


25i7 


2603 


2689 


2775 


in^ 


3o33 


3ii9 


33o5 


86 


5o5 


3291 


3377 


3463 


3549 


3635 


3721 


3893 


3979 


4o65 


86 


5o6 


4IDI 


4236 


4322 


4408 


4494 


4579 


4665 


4731 


4837 


4922 


86 


507 


5oo8 


5094 


5179 


5265 


535o 


5436 


5522 


5607 


5693 


5778 


86 


5o8 


5864 


5949 
68o3 


6o3o 


6120 


6206 


6291 


6376 


6462 


6547 


6633 


85 


5*9 


6718 


6888 


6974 


7059 


7144 


7229 


73i5 


7400 


7485 


85 


5io 


707570 
8421 


7655 


7740 


7826 


791 1 


8?46 


8081 


8166 


825i 


8336 


85 


5ii 


85o6 


8591 


8676 


8761 


8931 


9015 


9100 


9185 


85- 


5ii 


9270 


9355 


9440 


9524 


9609 


9694 


9779 


9863 


9948 


••33 


85 


5i3 


710117 


0202 


0287 


0371 


04d6 


0540 


062D 


0710 


0-94 


0879 


85 


5i4 


0963 

1807 


1048 


Il32 


1217 


i3oi 


1 385 


1470 


1 554 


i639 


1723 


84 i 


5i5 


1S92 


1976 


2060 


2144 


2229 


33i3 


23q7 
3238 


2481 


3566 


84 


5i6 


265o 


2734 


23i3 


2902 


29S6 


3o-o 


3i54 


3323 


3407 


84 


5id 


3491 


3575 


3659 


3742 


3826 


3910 


4833 


4078 


4162 


4346 


84 


433o 4414 


4497 


458i 


4665 


4749 


4916 


5ooo 


5o84 


84 


5i9 


5167 525i 

1 
1 I 


5335 
a 


5418 


55o2 


5586 


5669 


5753 


5836 


5930 


84 


N. 


3 


4 


5 


6 


7 


« 


9 


D. 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


520 


716003 


6087 


6170 


6254 


6337 


6421 


65o4 


6588 


6671 


6754 
7587 
8410 


83 


521 


6838 


6921 


7004 


7088 


8oo3 


7254 


7338 


7421 
8253 


7504 
8336 


83 


522 


■7671 
85o2 


?585 


6668 


7920 

8751 
9580 


8086 


8169 


83 


523 


8834 


8917 
9745 


9000 


9083 


9165 


9248 


83 


524 


9331 


9414 


9497 


9663 


9?^? 


99" 


9994 


••77 


83; 


525 


720159; 0242 


032D 


0407 
1233 


0490 


0573 


o655 


0738 


0821 j 0903 


U 


526 


0086 
1811 


1068 


ii5i 


i3i6 


1398 


1481 


i563 


1646 ' 1728 


62 


527 
528 


1893 


1975 


2o58 


2140 


2222 


23o5 


2387 


2469 ■ 2552 


82 


2634 


2716 


2798 


2881 


2963 


3045 


3127 
3948 


3209 


3291 


3374 


8a 


539 


3456 


3538 


3620 


3702 


3784 


3866 


4o3o 


4II2 


4194 


82 


53o 


724276 


4358 


4440 


4522 


4604 


4685 


4767 


4849 


4931 


5oi3 


82 


53 1 


5095 


5176 


5258 


5340 


5422 


55o3 


5585 


5667 


5748 


5830 


82 


532 


5912 


5993 


6075 


6i56 


6238 


6320 


6401 


6483 


6564 


6646 


8a 


533 


6727 


6809 
7623 


6890 


6972 


7053 


7134 


7216 


7297 


7379 


7460 


81 


534 


7541 


7704 


7780 


7866 


7948 


8029 


8110 


8191 


8273 


81 


535 


8354 


8435 


85i6 


8597 


8678 


8759 


8841 


8922 


9003 


9084 


81 


536 


9165 


9246 


9327 

•i36 


9408 


9489 


9570 


9651 


9782 


9813 


9893 


81 


537 
538 


9974 
730782 


••55 


•217 


•298 


•378 
u86 


•459 


•540 


•621 


•702 


81 


o863 


0944 


1024 


iio5 


1266 


1 347 


1428 


i5o8 


81 


539 


1589 


1669 


1730 


i83o 


1911 


1991 


2072 


2l52 


2233 


23i3 


81 


540 


732394 


2474 


2555 


2635 


2715 


2796 


2876 


2956 


3o37 


3117 


80 


541 


3197 


3278 


3358 


3438 


35i8 


3598 


3679 


^Ia9 
4560 


3839 


3919 


80 


542 


3099 


4079 


4160 


4240 


4320 


4400 


4480 


4640 


4720 


80 


543 


4800 


4880 


4960 


5o4o 


5l20 


5200 


5270 


5359 


5439 


55i9 


80 


544 


5599 


5679 


5739 


5838 


5918 


5998 


6078 


6i57 


6237 


6317 
7113 


80 


545 


6397 


6476 


6556 


6635 


6715 


6795 


6874 


6954 


7034 


80 


546 


7193 


7272 


7352 
8146 


7431 
8225 


75ii 


7590 

8384 


■7670 
8463 


854^ 


7829 
8622 


7908 


79 


547 


li] 


S^7 


83o5 


8701 


79 


548 


8860 


8939 


9018 


9097 


9177 


9256 


9335 


9414 


9493 


79 


549 


9672 


965i 


9731 


9810 


9889 


9968 


••47 


•126 


•2o5 


•284 


79 


55o 


74o363 


0442 


052I 


0600 


0678 


0757 


o836 


0915 


0994 


1073 


79 


55i 


Il52 


I23o 


1 309 


1388 


1467 


1546 


1624 


1703 


178? 


i860 


79 


552 


1939 


2018 


2096 
2882 


2175 


2254 


2332 


241 1 


2489 


2568 


2647 


75 
78 


553 


272D 
35io 


2804 


2961 


3o39 
3823 


3ii8 


3196 


3275 


3353 


343 1 


554 


3588 


3667 


3745 


3902 


3980 


4o58 


4i36 


42i5 


78 


555 


4293 


4371 
5i53 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


78 


556 


5075 


523i 


5309 


5387 


5465 


5543 


5621 


5699 


5777 


78 


557 
558 


5855 


5933 


6011 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


78 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7n? 


7256 


7334 
8110 


78 


559 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8o33 


78 


56o 


748188 


8266 


8343 


8421 


8498 


8576 


8653 


8731 


8808 


8885 


77 


56i 


8963 


9040 


9118 


9195 


9272 


935o 


9427 


9004 


9582 


9659 


77 


562 


9736 


9814 


9891 


9960 


••45 


•123 


•200 


•277 
1048 


•354 


•43 1 


77 


563 


75o5o8 


o586 


0663 


0740 


0817 


0894 


0971 


II25 


1202 


77 


564 


1279 
2048 


i356 


1433 


i5io 


i587 


1664 


1741 


1818 


1895 


197a 


77 


565 


2125 


2209 


2279 


2356 


2433 


2609 


2586 


2663 


2740 
35o6 


77 


566 


2816 


2893 


297© 


3o47 


3i23 


3200 


3277 


3353 


343o 


77 


567 
568 


3583 


366o 


3736 


38i3 


3889 


3966 


4041 


4119 

4883 


4195 


4272 


77 


4348 


4425 


45oi 


4578 


4654 


4730 


4807 


40CX) ' 5o36 


6 


569 


5lI2 


5189 


5265 


5341 


5417 


5494 


5570 


5646 


5722 


5799 


76 


570 


755875 
6636 


5951 


6027 
6788 
7548 
83o6 


6io3 


6180 


6256 


6332 


6408 


6484 


656o 


76 


571 


6712 


6864 


6940 


7016 


7092 

785i 


7168 


7244 
8oo3 


7320 
8079 


76 


572 


l^, 


7472 
8230 


7624 


7700 
8458 


7775 


7927 


76 


573 


8382 


8533 


8609 


8685 


8761 


8836 


76 


574 


8912 


8988 


9063 


9139 


9214 


9290 


9366 


9441 


9517 


9592 


76 


575 


9668 9743 


9819 


9894 


9970 


••45 


•121 


•196 


•272 


•347 


75 


576 


760422 


0498 

I25l 


0573 


0649 


0724 


0799 


0875 


0930 


1025 


IIOI 


75 


m 


1176 


i326 


1402 


1477 


i552 


1627 
2378 


1702 


1778 


i853 


75 


1928 
2679 


2003 


2078 


2i53 


2228 


23o3 


2453 


2529 


2604 


75 


579 


2754 


2829 


2904 


2978 


3o53 


3i28 


32o3 


3278 


3353 


75 


N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

i 



LO 



A TABLE OF LOGARITHMS FROM I TO 10,000. 



N. 
58o 





I I 2 


3 


4 5^617 


1 ^ 


9 


"bTI 
75 


763428 


35o3 


3578 


3653 


3727 38o2 ' 3877 i 8952 


! 4027 1 4101 


58i 


4176 


425i 


4826 


4400 


4475 455o ■ 4624 4699 


! 4774 1 4848 


75 


582 


4923 


4998 


5072 


5i47 


5221 5296 5370 5443 


: 5520 ' 5594 
6264 1 6338 


1 75 


583 


5669 
6413 


5743 


58i8 


5892 


5966 


6041 


; 6ii5 1 6190 
6859 1 6988 


'' u 


584 


6487 


6562 


6686 


6710 


6785 


7007 , 7082 


'. -4 


^85 


7i56 


7i3o 


1 7^04 


7879 


7458 

1 8x94 

8984 


7527 


7601 
8843 


7675 


7749 i 7«23 


,4 


586 


7898 


7972 


i 8046 


B120 


8268 


8416 


! 8490 


1 8564 


■ 74 


587 


8638 


8712 


8786 


8860 


9008 


9082 


9i56 


9280 


g3o3 


74 


588 


9377 


9451 


1 9525 


9599 


9673 


9746 


9820 


1 9894 
0681 


9968 


! ••42 


74 


589 


770II3 


0189 


0263 


o336 


0410 


0484 


0557 


0705 


i 0778 


74 


590 


770852 


0926 


0999 


1078 


1 146 


1220 


1293 


1867 


1440 


i5i4 


74 


591 


1587 


1661 


1734 


1808 


i88i 


1955 


2028 


2102 


2175 


2248 


73 


592 


2322 


2395 


2468 


2542 


26i5 


2688 


2762 


2885 


2908 


2981 


73 


593 


3o55 


3128 


8201 


8274 


3848 


8421 


3494 


3567 


8640 


3718 


' 73 


594 


3786 


3860 


3988 


4006 


4079 


4i52 


4225 


4298 


4371 


4444 


73 


595 


4517 


4590 


4668 


4786 


4809 


4882 


4955 


5o28 


5ioo 


5178 


73 


596 


5246 


5319 


5392 


5465 


5588 


56io 


5683 


5756 


5829 


5902 


73 


597 


5974 


6047 


6120 


6198 


6265 


6338 


641 1 


6488 


6556 


6629 


73 


598 


6701 


6774 


6846 


6919 


6992 


7064 


7187 


7209 


7282 
8006 


7354 
8079 


73 


599 


7427 


7499 


7572 


7644 


7717 


7789 


7862 


7984 


72 


600 


778i5i 


8224 


8296 


8868 


8441 


85i3 


8585 


8658 


8780 


8802 


72 


601 


8874 


^1P 


9019 


9091 


9168 


9286 


9808 


9880 


9452 


9524 


72 


602 


9596 


9669 


9741 


9818 


9885 


9957 


••29 


•lOl 


•173 


•245 


72 


6o3 


780817 


0889 


0461 


o588 


o6o5 


0677 


0749 


0821 


0898 


0965 


72 


604 


loS-) 


1 109 


1181 


1258 


1824 


1896 


1468 


1540 


1612 


1684 


72 


6o5 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 


2829 


2401 


72 


606 


2473 


2544 


2616 


2688 


2739 


2881 


2902 


2974 


8046 


8117 


72 


607 


3189 


3260 


3882 


8408 


8473 


3546 


36i8 


8689 


8761 


3832 


71 


608 


3904 


3975 


4046 


4118 


4189 


4261 


4832 


44o3 


f^r 


4546 


71 


609 


4617 


4689 


4760 


4881 


4902 


4974 


5o45 


5n6 


5187 


5259 


71 


610 


785330 


5401 


5472 


5543 


56i5 


5686 


5757 


5828 


5899 


5970 


11 


611 


6041 


6112 


6188. 


6254 


6325 


6896 


6467 


6588 


6609 


6680 


11 


612 


6751 


6822 


6898 


6964 


7o85 


7106 


7177 


7248 


7819 


7890 


7» 


6i3 


7460 
8168 


7531 


7602 


7673 


7744 


7815 


7885 


7956 


8027 


8098 


711 


614 


8289 


83io 


838 1 


8451 


8522 


8598 


8663 


8784 


88c4 


7^ 


6i5 


8875 


8946 


9016 


9087 


9157 


9228 


9299 


9869 


9440 


95io 


71 


616 


9581 


9631 


9722 


9792 


9868 


9988 


•••4 


••74 


•144 


•2l5 


70 


618 


790285 


o356 


0426 


0496 


0367 


0687 


0707 


0778 


0848 


0918 


70 


0988 


1059 


1 1 29 


1 199 


1269 


1840 


1410 


1480 


i55o 


1620 


70 


619 


1691 


1761 


1881 


1901 


1971 


2041 


2111 


2l8l 


2252 


2822 


70 


620 


792392 


2462 


2582 


2602 


2672 


2742 


2812 


2882 


2952 


3o22 


70 


621 


3092 


8162 


3281 


33oi 


8871 


3441 


35ii 


358i 


365i 


8721 


70 


622 


3790 


386o 


8980 


4000 


4070 


4189 


4209 


4279 


4349 


4418 


70 


623 


4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5043 


5ii5 


70 


624 


5i85 


5254 


5824 


5393 


5468 


5582 


56o2 


5672 


5741 


58ii 


^ 


6a5 


588o 


5949 


6010 


6088 


6i58 


6227 


6297 , 6366 


6436 


65o5 


626 


6574 


6644 


6713 


6782 


6852 


6921 


699c 


7060 


7129 


7198 


69 


^^1 
6sd 


7268 


7337 


7406 


747^ 


7545 


7614 


7688 
6374 


7752 


7821 7890 


69 


7960, 8029 


8098 


816- 


8236 i 


88o5 


8443 


85i3 1 8582 


69 


629 


865i 8720 


8789 


8858 


8927 , 


8996 


9065 


9184 


9208 1 9272 


69I 


63o 


799341 


9409 
0098 


9478 


9547 


9616 . 


9685 


9754 


9828 


9892 1 9961 : 


^i 


63i 


800029 


0167 


0286 


o3o5 


0878 


0442 


o5ii 


03b0 ; 0648 


69 


632 


0717: 0786 


o854 


0928 


0992 , 


1061 


1 1 29 


1 198 


1266 j i885 


691 


633 


i4o4i 1472 


1 541 


1609 
2295 ! 


1678 [ 


1747 


i8i5 


1884 


1952 2021 ! 


69 


634 


2089 2 1 58 


2226 


2868 1 


2482 ! 


i5oo 


2568 


2687 


2705 1 


^ 


635 


2774 2842 


2qio 


2979 j 3o47 


3ii6 


3i84 


3252 


3821 


8889 1 


68 


636 


3457 


3525 35q4 


3662 i 3780 


8708 


8867 


8985 


4008 


4071 1 


68 


637 
638 


4139 


4208 


4276 


4344 4412 


4480 ; 


4548 i 4616 


4685 4753 : 


68 


4821 


4889 


4q57 i 


5o25 5098 


5i6i 


5220 


5297 ; 


5365 5433 i 


68 


639 


55oi 


5569 5637 

i 1 


5705 5773 5841 


5908 


5976 


6044 1 61 1 2 


68 
D 


N. 


I 


3 


3 


4 5 1 6 


7 ! 


8 


__9j 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



11 



N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D. ; 


64o 


806180 


6248 


63i6 


6384 


645 1 


65i9 


6587 


6655 


6723 


6790 


68 


641 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7882 


7400 


8143 


68 


642 


7535 


7603 


1670 
8346 


7738 
8414 


7806 


7873 


7941 


8008 


0076 


6d) 


' 643 


8211 


8279 
8953 


8481 


8549 


8616 


8684 


8751 


8818 


67] 


i 644 


8886 


9021 


9088 


9i56 


9223 


9290 


9358 


9425 


9492 


67! 


1 645 


9560 


9627 


9694 


9762 


9829 


9896 


9964 


••3 1 


••98 1 •i65 


67. 


! 646 


810233 


o3oo 


0867 


0484 


o5oi 


0569 


0686 


0708 


0770 


0887 
i5o8 


671 


1 647 


0904 


0971 


1089 


U06 


1178 


1240 


1807 


1874 


1441 


67' 


i 648 


1575 


1642 


1709 


1776 


1843 


1910 
2679 


1977 


2044 


2111 


2178 


6? 


! 649 


2245 


23l2 


2379 


2445 


25l2 


2646 


2718 


2780 


2847 


67 


65o 


812913 


2980 


3047 


3ii4 


3i8i 


3247 


83i4 


338i 


3448 


35i4 


67 


65 1 


3o8i 


3648 


3714 


3781 


3848 


8914 


8981 


4048 


4114 


4181 


67 


652 


4248 


43i4 


438i 


4447 


45i4 


458 1 


4647 


4714 4780 


4847 


67 


653 


4913 


4980 


5o46 


5ii8 


5170 


5246 


58i2 


5378 


5445 


55ii 


66 


654 


5D78 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 


66 


655 


6141 


63o8 


6374 


6440 


65o6 


6578 


6689 


6705 
7367 
8028 


6771 


6838 


66 


656 


6904 


6970 


7o36 


7102 


7169 


7285 


7801 


7483 


7499 


66 


657 


7565 
8226 


7631 
6292 


7698 
8358 


7764 


7880 


7896 
8556 


7962 


8094 


8160 


66 


658 


8424 


8490 


8622 


8688 


8754 


8820 


66 


659 


8885 


8951 


9017 


9083 


9149 


92i5 


9281 


9846 


9412 


9478 


66 


660 


819544 


9610 


9676 


9741 


9807 


9873 


n^i 


•••4 


••70 


•i36 


66 


! 661 


820201 


0267 


o333 


0899 


0464 


o53o 


oSoS 


0661 


0727 


0792 


66 


662 


o858 


0924 


0989 


I ODD 


1 1 20 


1 186 


I25l 


1817 


1882 


1448 


66 


663 


i5i4 


1579 


1645 


17IO 


1775 


1 841 


1906 


1972 


2087 


2io3 


65 


664 


2168 


2233 


2299 


2364 


243o 


2495 


256o 


2626 


2691 


2756 


65 


665 


2822 


2887 


2902 


3oi8 


8088 


8148 


8218 


8279 


3844 


3409 


65 


666 


3474 


3539 


36o5 


3670 


8785 


38oo 


3865 


3980 


8996 


406 1 


65 


667 
668 


4126 


4I9I 


4256 


4821 


4886 


445 1 


45i6 


458 1 


4646 


471 1 


65 


4776 


4841 


4906 


4971 


5o36 


5ioi 


5i66 


5281 


5296 


536i 


63 


669 


5426 


5491 


5556 


5621 


5686 


5751 


58 J. 5 


588o 


5945 


6010 


65 


670 


826075 


6140 


6204 


6269 


6884 


6899 


6464 


6528 


6598 


6658 


65 


671 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7805 


65 


672 


7369 


7434 


7499 


7068 


7628 


7692 


7757 


7821 


7886 


7951 


65 


673 


8oi5 


8080 


8144 


8209 


8273 


8888 


8402 


8467 


858 1 


8595 


64 


674 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9289 


64 


675 


9804 


9368 


9432 


9497 


9561 


9625 


9690 


9754 


9818 


9882 


64 


676 


9947 


••11 


••75 


•189 


•204 


•268 


•832 


•896 


•460 


•525 


64 


t]l 


83o589 


0653 


0717 


0781 


0845 


0909 


0978 


1087 


1102 


1 166 


64 


i^23o 


1294 


i358 


1422 


i486 


i55o 


1614 


1678 


1742 


1806 


6a, 


679 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2817 


2881 


2445 


64 


680 


832309 


2573 


2687 


2700 


2764 


2828 


2892 


2956 


8020 


3o83 


6A 


681 


3 147 


32II 


8275 


3388 


8402 


3466 


853o 


3598 


3657 


3721 


64 


682 


3784 


3848 


3912 


8975 


4089 


4108 


4166 


4280 


4294 


4357 


64 


683 


4421 


4484 


4548 


4611 


4675 


4789 


4802 


4866 


4929 


4008 


64 


684 


5o56 


5l20 


5i83 


5247 


58io 


5378 


5437 


55oc 


5564 1 5627 


63 


685 


5691 


5754 


5817 


588 1 


5944 


6007 


607; 


6184 


6197 


6261 


63 


686 


6324 


6387 


645 1 


65i4 


6577 


6641 


6704 


6767 


6880 


6894 


63 


687 


6957 
7588 


7020 


7088 


7146 


7210 


7278 


7886 


7899 


7462 


7525 


63 


688 


7652 


77i5 


77-'8 
8408 


7841 


7904 
8534 


7967 


8080 


8098 


8i56 


63 


I 689 


8219 


8282 


8345 


8471 


8597 


8660 


8728 


8786 


63 


1 690 


83884Q 


8912 


8975 


9088 


9101 


9164 


9^?I 


9289 


9852 


041 5 


63 


691 


9478 


9541 


9604 


9667 


9729 


9792 


9855 


9918 


9981 j ••43 


63 


692 


840106 


0169 


0282 


0294 


o857 


0420 


0482 


o545 


0608 ; 0671 


63 


693 


0733 


0796 


0850 


0921 


0984 


1046 


1 109 


1172 


1284 1297 


63 


1 694 


1359 
1985 


1422 


1485 


1 547 


1610 


1672 


1735 


1797 


i860 ! 1922 


63 


1 695 


2047 


2110 


2172 


2235 


2297 


2860 


2422 


2484 2547 


63 


' 696 


2609 


2672 


2734 


2796 


2859 


1921 


2988 


8046 


8108 8170 


6a 


S 


3233 


3295 


3857 


3420 


3482 


3544 


36o6 


8669 


8781 8798 


6a 


3855 


3918 


8980 


4042 


4104 


4166 


4229 


4291 


4358 1 441 5 


6a 


699 
N. 


4477 


4539 


4601 


4664 


4726 


4788 


485o 


4912 


4974 ' 5o36 


63 
0, 





I 


2 1 3 


4 


5 


6 


7 


■ 
8 


9 



L2 



A TABLE OF L0GAEITHM3 FROM 1 TO 10,000. 



N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


1)7 


; 700 


845098 


5i6o 


5222 


5284 


5346 


5408 


5470 


5532 


5594 


5656 


62 


701 


5718 


5780 


5842 


5904 


5966 
6585 


6028 


6090 


6i5i 


6213 1 6275 


62 


702 


6337 


6399 


6461 


6523 


6646 


6708 


6770 


6832 ] 6894 


62 


703 


6955 


7017 


1 7079 ' 7141 


7202 


7264 


7326 


7388 


7449 ' 75ii 


62 


704 


7073 


7634 


7396 ' --ToS 


7819 


7881 


7943 


8004 


8066 


^128 


62 


! 705 


8189 


825i 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


1 3743 


6a 


706 


88o5 


8866 


8928 


8989 


905 1 


9112 


9174 


9235 


9297 


1 p358 


61 


]o^ 


9419 


9481 


9542 


9604 


9665 


9726 


9788 


9849 


991 1 
o524 


9972 


61 


85oo33 


0095 


oi56 


0217 


0279 


o34o 


0401 


0462 


0585 


61 


709 


0646 


0707 


0769 


o83o 


0891 


0962 j 1014 


1075 


ii36 


1 197 


61 


710 


851258 


l320 


i38i 


1442 


i5o3 


1 564 


1625 


1686 


1747 


1809 


61 


711 


1870 
2400 


I93I 


1992 


2o53 


2114 


2175 


2236 


2297 


2358 


2419 


61 


'''^ 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


61 


713 


3090 


3i5o 


32II 


32T2 


3333 


3394 


3455 


35i6 


3577 


3637 


61 


714 


3698 


3759 


3820 


388i 


3941 


4002 


4o63 


4124 


4i85 


4245 


61 


715 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 


61 


716 


4913 


4974 


5o34 


5095 


5i56 


5216 


5277 


5337 


5398 


5459 


61 


718 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6oo3 


6064 


61 


6124 


6i85 


6245 


63o6 


6366 


6427 


6487 


6548 


6608 


6668 


60 


719 


6729 


6789 


685o 


6910 


6970 


7o3i 


7091 


7i52 


7212 


7272 


60 


720 


857332 


7393 


7453 


75i3 
8116 


7^74 


7634 


7694 


7755 


78i5 
8417 
9018 


7875 


60 


721 


7935 


7995 


8o56 


8176 


8236 


8297 


8357 


8477 


60 


722 


8537 


8697 
9198 


8657 


8718 


8778 


8838 


8898 


8958 


9078 


60 


723 


9i38 


9258 


9318 


9370 


9439 


9499 


9559 


9619 


9679 


60 


724 


9739 


9799 


9859 


9918 


9978 


••38 


••98 


•i58 


•218 


•278 


60 


725 


86o338 


0398 


0458 


o5i8 


0678 


0637 


0697 


0757 


0817 
i4i5 


0877 


60 


726 


0937 


0996 


io56 


1116 


1176 


1236 


1295 


i355 


1475 


60 


727 
728 


1 534 


1594 


1654 


1714 


1773 


1 833 


1893 


1952 


2012 


2072 


60 


2l3l 


2191 


225l 


23lO 


2370 


243o 


2489 


2549 


2608 


2668 


60 


729 


2728 


2787 


2847 


2906 


2966 


3o25 


3o85 


3i44 


3204 


3263 


60 1 


730 


863323 


3382 


3442 


35oi 


356i 


3620 


368o 


3739 


3799 


3858 


59 


71 


3917 


3977 


4o36 


4096 


4i55 


4214 


4274 


4333 


4392 


4452 


59; 


ll^ 


43II 


4570 


463o 


4689 


4748 


4808 


4867 


4926 


4985 


5o45 


59 


733 


5io4 


5i63 


5222 


5282 


5341 


5400 


5459 


55i9 


5578 


5637 


59I 


734 


56q6 


5755 


58i4 


5874 


5933 


5992 


6o5i 


6110 


6169 


6228 


59 1 


'^ii 


6287 
6878 


6346 


64o5 


6465 


6524 


6583 


6642 


6701 


6760 


6819 


59 j 


736 


6937 


6qq6 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


59 


]ll 


7467 


7026 


J 


7644 


7703 


7762 


7821 


7880 


7939 


7998 


59 


8o56 


8ii5 


8174 


8233 


8292 


835o 


8409 


8468 


8527 


8586 


59 


739 


8644 


8703 


8762 


8821 


8879 


8938 


8997 


9o56 


9114 


9173 


59 


740 


869232 


9290 


9349 


9408 


9466 


9525 


9584 


9642 


9701 


9760 


^ 


741 


9818 


9877 


9935 


9994 


••53 


•hi 


•170 


•228 


•287 


•345 


59 


742 


870404 


0462 


052I 


0579 


o638 


0696 


0755 


o8i3 


0872 


0930 
i5i5 


58 


743 


0989 


1047 


1 106 


1 164 


1223 


1281 


1 339 


1398 


1456 


58 


744 


1573 


i63i 


1690 


1748 


1806 


1 865 


1923 


1981 


2040 


2098 


58 


745 


2i56 


22l5 


2273 


233i 


2389 


2448 


25o6 


2564 


2622 


2681 


58 


746 


2739 


2797 


2855 


2913 


2972 


3o3o 


3o88 : 3146 1 


3204 


3262 


58 


747 3321 


3379 


3437 
4018 


3495 


3553 


36ii 3669 : 2727 I 


3785 


3844 


58 


748 


3902 


3960 


4076 


4i34 


4192 


425o 43o8 


4366 


4424 i 


58 


749 


4482 


4540 


4598 


4656 


4714 


4772 


4830 4888 


4945 


5oo3 


58 


75o 


875061 


5ii9 


5i77 


5235 


5293 


535i 


5409 1 5466 


5524 


5582 


58 


75i 


5640 


5698 


5756 


58i3 


5871 


5929 


5987 


6045 


6102 1 


6i6o 


58 


752 


6218 


6276 


6333 


6391 


6449 


65o7 


6564 


6622 


6680 1 


6737 


58 


753 


6795 


6853 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


7314 


58 


754 


7371! 


7429 


7487 


7D44 


7602 


7659 


7717 7774 


7832 


7889 
8464 


58 


755 


7947 


8004 ' 


8062 ' 


8119 


8177 


8234 


8292 ; 8349 


8407 


?7 


756 


8522 


8579 I 


8637 


8694 


8752 


8809 


8866 1 8924 


8981 


90 J9 


^7 


^h 


9096 


9153 9211 9268 


93a5 


9383 


9440 


9497 


9555 


9612 


i^ 


758 


9669 


9726 9784 9841 


9898 


9o56 
o528 


••i3 


••70 


•127 


•i85 1 


57 


759 


880242 


0299 o356 041 3 


0471 


o585 


0642 


0699 


0756 1 


^7 





I 


3 3 


4 


5 


6 


7 


8 


9 


D. 



A TABLE OP LOGARITHMS FROM 1 TO 10,000. 



18 



N. 
I 760 





I 


2 


3 


1 ^ 


I 5 


1 
6 


7 


8 


9 


D. 
57 


880814 


0871 


0928 


0985 


1042 


1099 


ii56 


12l3 


1271 


1828 


ZJ' 


r385 


1442 


1499 


1 556 


1618 


1670 


1727 


1784 


1 841 


1898 


57 


' ^? 


io55 

2D25 


2012 


2069 


2126 


2188 


2240 


2297 


2354 


241 1 


2468 


57 


1 763 


258i 


2638 


2695 


2752 


2809 


2866 


2928 


2980 


8087 


57 


764 


8098 


3i5o 


8207 


8264 


8321 


3877 


8434 


8491 


8348 


36o5 


57 


i 765 


1 366 1 


8718 


3775 


8882 


8888 


8945 


4002 


4069 


4ii5 


4172 


57 


/66 


4229 


4285 


4342 


4899 


4455 


45i2 


4569 
5i85 


4625 


4682 


4780 


571 


1 767 
i 76B 


4795 
536 1 


4852 


4909 


4965 
5531 


5022 


5078 


5192 


5248 i 53o5 


57 i 


5418 


5474 


5587 


5644 


570c 


6821 


58i3 5870 


U 


1 769 


5926 


5983 


6089 


6096 


6i52 


6209 


6265 


6878 1 6484 


i 770 


886491 
7064 


6547 


6604 


6660 


6716 


6773 


6829 


6885 


6942 


6998 


56 


' 771 


7111 


7167 


7223 


- 7280 


7^^ 


73^2 


7449 


75o5 


756i 


56 


772 


7617 
8179 


7674 


7780 
8292 


ViA^ 


7842 
8404 


7898 


llib 


801 1 


8067 


8128 


56 


773 


8236 


8460 


8573 


8629 


8685 


56 


774 


8741 


8797 


8858 


8909 


8965 


9021 


9077 


9184 


9190 


9246 


56 


•,75 


9802 


9358 


9414 


9470 


9526 


9582 


9688 


9694 


9730 


9806 


56 


776 


9862 


9918 


9974 


••3o 


••86 


•141 


•197 


•253 


•809 


•865 


56 


777 
77S 


890421 


0477 
io85 


o588 


o589 


0645 


0700 


0756 


0812 


0868 


0924 


56 


0980 


1091 


1 147 
1705 


1208 


1209 


i3i4 


1870 


1426 


1482 


56 


779 


1537 


1598 


1649 


1760 


1816 


1872 


1928 


1988 


2089 


56 


780 


^'6t 


2i5o 


2206 


2262 


2817 


2878 


2429 


2484 


2540 


2595 


56 


7?' 


2707 


2762 


2818 


2878 


2929 


2985 


8040 


8096 


8i5i 


56 


7?? 


3207 


8262 


8818 


3378 


8429 


3484 


3D40 


8505 


865i 


8706 


56 


783 


3762 


8817 


8878 


3928 


8984 


4089 


4094 


4i5o 


42o5 


4261 


55 


784 


43i6 


4871 


4427 


4482 


4538 


4598 


4648 


4704 


4759 


4814 


55 


7?^ 


4870 


4925 


4980 


5o86 


5091 


5146 


5201 


5257 


58i2 


5867 


55 


! 786 


5423 


5478 


5d83 


5588 


5644 


5699 


5754 


5809 


5864 


5920 


55 


i 7^7 


5975 
6526 


6o3o 


6o85 


6140 


619* 


6201 


6806 


6861 


6416 


6471 


55 


1 788 


658i 


6686 


6692 


6747 


6802 


6857 


6912 


6967 


7022 


55 


; 789 


7077 


7182 


7187 


7242 


7297 


7352 


7407 


7462 


7517 


7572 


55 


1 790 


897627 


7682 
8281 


7737 
8286 


7792 


7847 
8896 


7902 


7o5j 
85o6 


8012 


8067 
86i5 


8122 


55 


i 791 


8176 


8841 


8451 


856i 


8670 


55 


792 


8725 


8780 


8885 


8800 


8944 


8999 


9054 


9109 


9164 


9218 


55 


M93 


9273 


9828 


9888 


9^^I 


9492 


9547 


9602 


9656 


97II 


9766 


55 


794 


9821 


9875 


9980 


9985 


••89 


••94 


•149 


•203 


•258 


•3l2 


55 


i 795 


900867 


0422 


0476 


o53i 


o586 


0640 


0695 


0749 


0804 


0859 


55 


796 


0913 


0968 


1022 


1077 


ii3i 


1 186 


1240 


1293 


1849 


1404 


55 


]'^ 


1458 


i5i8 


1 567 


1622 


1676 


1781 


1785 


1840 


1894 


1948 


54 


2003 


2057 


2112 


2i66 


2221 


2275 


2829 


2884 


2488 


2492 
8o36 


54 


799 


2547 


2601 


2655 


2710 


2764 


2818 


2873 


2927 


2981 


54 


Boo 


908090 


8144 


8199 


3258 


8807 


3861 


8416 


3470 


3524 


3578 


54 


3oi 


3633 


3687 


8741 


3795 


8849 


3904 


8953 


4012 


4066 


4120 


54 


3o2 


4174 


4229 


4288 


^t^l 


4891 


4445 


4499 


4558 


4607 
5i48 


4661 


54 


3o3 


4716 


4770 


4824 


4878 


4982 


4986 


5o4o 


5094 


5202 


54 


804 


5256 


58io 


5364 


5418 


5472 


5526 


558o 


5634 


5688 


5742 


541 


8o5 


5796 
6335 


585o 


5904 5958 1 


6on 


6066 


6119 


6178 


6227 


6281 


54 


806 


6889 


6443 


6497 


655i 


6604 


6658 


6712 


6766 


5820 


54 


807 
808 


6874 


6927 


6981 


7089 


7143 


7196 
7734 


7250 7804 


7858 


54 


741 1 


7465 


7319 


1578 
8no 


7626 
8168 


7680 
8217 


7787 ! 7841 
8824 8878 


7805 
8481 


54 


809 


7949 


8002 


8o56 


8270 


54 


810 


908485 


8539 


8592 


8646 


8609 8753 
9235 9289 


8807 


8860 8914 


8961 


54 


811 


9021 


9074 


9128 


9181 


9842 


9896 9449 


93o3 


54 


812 


9556 


9610 


9668 


9716 


9770 9823 


9877 


9980 


9984 


••37 


53 


8i3 


910091 


0144 


0197 
0781 


025l 


0804 o858 


041 1 


0464 


o5i8 


0571 


53 


814 


0624 


0678 


0784 


o838 


0891 


0944 


0998 


io5i 


1 104 


53 


8i5 


ii58 


1211 


1264 


i3i7 


187. 


1424 


1477 


i53o 


i584 


1687 


53 


816 


1690 


1743 


1797 


i85o 


1908 


1956 


2009 


2o63 


2116 


2169 


53 


^'7 
818 


2221 


2275 


2828 


2881 


2485 


2488 


2541 


2594 


2647 
8178 


2700 


53 


2753 


2806 


2859 


2918 


2066 


8019 


8072 


3i25 


8281 


53 


819 


3284 


3337 


8890 


8443 , 3496 


3549 


8602 


3655 


8708 


3761 


53 


H. 





I 


3 


3 i 4 


5 6 


7 


8 


9 


D. 



14 



A TABLE OF LOGARITHMS FROM I TO 10,000. 



N. 



820 
831 
822 
823 
824 
825 
826 
827 
828 
829 

83o 
83 1 
832 
833 
834 
835 
836 
837 
838 
839 

840 
841 
842 
843 
844 
845 
846 
847 
848 
849 

85o 
85 1 
852 
853 
854 
855 
856 
857 
858 
859 

860 
861 
862 
863 
864 
865 
866 
861 
868 
869 

870 
871 
872 
873 
8:4 
875 
876 
877 
878 

879 
N. 



9i38i4 
4343 i 

48721 
54001 
59271 
6454 
6980; 
7D06' 
8o3o 
8555 

919073: 

9601 ; 
920123! 
0645! 
1166 
1686 
2206 
2725 

3244 
3762 

924279 
4796 
53i2 
5828 
6342 
6857 
7370 
7883 
8396 
8908 

929419! 
9930 

930440 1 
0949 1 
I458| 
1966 
2474 
2981 
3487; 
3993 

9344981 
5oo3 
5507 
601 1 
65i4 
7016 
7518 
8019 
8520 
9020 

939519 

940018 

o5i6 

!0I4 

i5ii 

2008 
25o4 
3 000 
3495 
3989 



3867 
4396 
4925 
5453 
5980 
6507 
7033 
7558 
8o83 
8607 

9i3o 
9653 
0176 
0697 
1218 
1738 

2258 

2777 
3296 
38i4 

433 1 
4848 
5364 
5879 
6394 
6908 
7422 
7935 

8447 
8959 

9470 
9981 
0491 
1000 
1 509 
2017 

2524 

3o3i 

3538 
4044 

4549 
5o54 
5558 
6061 
6564 
7066 
7568 
8069 
8570 
9070 

9569 
0068 
o566 
1064 
i56i 
2o58 
2554 
3o49 
3544 
4o38 



2 


3 


3920 


3973 


4449 


4302 


4977 


5o3o 


55o5 


5558 


6o33 


6o85 


6559 

7085 


6612 


7i38 


761 1 
8i35 


7663 
8188 


8659 


8712 



9183 

9706 
0228 

0749 

1270 

1790 

23lO 

2829 
3348 

3865 

4383 
4899 
5413 
5931 
6445 
6959 
7473 
7986 
8498 
9010 

9521 

••32 

o542 
io5i 
i56o 
2068 
2575 
3082 
3589 
4094 

4599 
5io4 
56o8 
6111 
6614 
7117 
7618 
8119 
8620 
9120 

9619 

Olio 

0616 i 

III4 

I6II 

2107 

26o3 

3099 

3593 

4088 



4026 
4555 
5o83 
56ii 
6i38 
6664 
7190 
7716 
8240 
8764 

9235 I 5287 

9758 '9810 

0280 I o332 

0801 I o853 



l322 

1842 
2362 

2881 
3399 
3917 

4434 
4951 
5467 
5982 

6497 

701 1 

7524 

8037 

8549 
9061 

9572 
••83 
0592 

II02 
161O 
2I18 
2626 

3i33 
3639 
4145 

4650 
5i54 
5658 
6162 
6665 

7167 
7668 
8169 
8670 
9170 

9660 
0160 ■ 
0666 
:i63 
660 
2157 
2653 
3148 
3643 
4137 



1374 
1894 
2414 
2933 
345i 
3969 

4486 
5oo3 
55i8 
6o34 
6548 
7062 
7576 
8088 
8601 
9112 

9623 
•i34 
0643 
ii53 
1661 
2169 
2677 
3i83 
3690 
4195 

4700 
52o5 
5709 
6212 
6715 
7217 
7718 
8219 
8720 
9220 

971Q 
0218 
0716 

-2l3 

I7I0 
2207 
2702 

3198 

36o2 
4186 



4070 
4608 
5i36 
5664 
6191 
6717 
7243 
7^68 
8293 
8816 

9340 
9862 
o384 
0906 
1426 
1946 
2466 
2985 
3do3 
4021 

4538 
5o54 
5570 
6o85 
6600 
7114 
7627 
*8i4o 
8652 
9163 

9674 
•i85 
0694 
1204 
1712 
2220 
2727 
3234 
3740 
4246 

4751 
5255 
5759 
6262 
6765 
7267 
7769 
8269 
8770 
9270 

9769 
0267 
0765 
1263 
1760 

2256 

2752 

3247 
3742 

4236 



4i32 
4660 
5189 
5716 
6243 
6770 
7295 
7820 
8345 
8869 

9392 
9914 
0436 
0958 
1478 
1998 
25i8 
3o37 
3555 
4072 

4589 
5io6 
5621 
6137 
665i 
7165 
7678 

8703 
9215 

9725 
•236 
0745 
1254 
1763 
2271 
2778 
3285 
3791 
429b 

4801 
53o6 
5809 
63i3 
68i5 
7317 
7819 
8320 
8820 
9320 

9819 
o3i7 
o8i5 
i3i3 
1809 
23o6 
2801 
3297 

4285 



4184 
4713 
5241 
5769 
6296 
6822 
7348 
7873 
8397 
8921 

9444 
9967 
9489 

lOIO 

i53o 
2o5o 
2570 
3089 
3607 
4124 

4641 
5i57 
5673 
6188 
6702 
7216 
7730 
8242 
8754 
9266 

9776 
•287 
0796 

l303 

I8I4 

2322 
2829 

3333 
3841 
4347 

4852 
5356 
5860 
6363 
6865 
7367 
7869 
$370 
8870 
9369 

9869 
o367 
o865 
i362 
1859 
2355 
2851 
3346 
3841 
4335 



8 



D. 



4237 I 4290 
4766 4819 
5294 1 5347 
5822 I 5875 
6349 I 6401 
6875 6927 
7400 745i 
7925 7978 
8450 ! 85o2 
8973 I 9026 



9496 
••19 
o54i 
1062 
i582 
2102 
2622 
3i4o 
3658 
4176 

4693 
5209 
5725 
6240 
6754 
7268 
7781 
8293 
88o5 
9317 

9827 
•338 
0847 
i356 
i865 
2372 
2879 



4902 
5406 
5910 
6413 
6916 



9549 
••71 
0593 
1114 
1634 
2i54 
2674 
3192 
3710 
4228 

4744 
5261 
5776 
6291 
68o5 
7319 
7832 
8345 
8857 
9368 

9?79 
•389 

0898 

1407 

1913 

2423 

2930 



3386 , 3437 
3392 J943 
4397 4448 



4953 

5457 
5960 
6463 
6966 



7418 i 7468 
7919 I 7969 
8420 I 8470 
8920 I 8970 
9419 9469 



9918 

0417 
0915 
1412 
1909 
24o5 
2901 
3396 
3890 
4384 

8 



9968 
0467 
0964 
1462 
1958 
2455 
2950 
3445 
3939 
4433 



53' 

53 

53 

53 

53 

:>3 

53 

5a 

52 
52 



5i 
5i 
5i 
5i 
5i 
5i 
5i 
5i 
5i 



^']±} 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



15 



N. 





I 


3 


3 


4 


5 


6 1 7 


8 


9 


49 


88o 


944483 


4532 1 458 I 


463 1 


4680 1 4729 


4779 ! 4828 


4877 


4927 


88i 


4976 


5o25 1 5o74 


5i24 


5173 


5222 


5272 


5321 


5370 


5419 


49 


882 


5469: 55i8 1 5567 


56i6 


5665 


5715 


5764 


58i3 


5862 


5912 


49 


883 


5961 


6010 1 6o59 


6108 


6157 


6207 


6256 


63o5 


6354 


6403 


49 


884 


6452 


65oi 


655i 


6600 


6649 


6698 


6747 


6706 


6845 


6894 


49 


885 i 6y43 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 


7385 


49 


m. 


7434 


7483 


i532 
8022 


75§I 

8070 


7630 


7679 


7728 


1 7777 


7826 


7875 


49 


' 887 


7924 
841 3 


7973 


81 19 


8168 


8217 


[ 8266 ' 83 1 5 1 8364 


49 


338 


8462 85 1 1 


856o 


8609 


8657 


8706 


8755 i 8804 


S853 


49 


1 889 


8902 


8951 


8999 


9048 


9097 


9146 i 9195 


9244 


9292 


9341 


49 


890 


949390 


9439 


9488 


953w 


; j585 


9634 


9683 


9731 


9780 


9829 


49 


891 


.9?]? 


9926 


9975 


••24 


-73 


•121 


•170 


•219 


•267 


•3i6 


49 


892 


95o36d 


0414 


0462 


o5ii 


o56o 


0608 


0657 


0706 


0754 


o8o3 


49 


893 : o«5i 


oqoo 


0949 


0997 


1046 


1095 


1 143 


1192 


1240 


1289 


49 


894 


1338 


1 386 


1436 


1483 


i532 


i58o 


1629 


1677 


1726 


1775 


49 


895 


1823 


1872 


1020 


1969 


2017 


2066 


2114 


2i63 


2211 


2260 


48 


896 


23o8 


2356 


24o5 


2453 


25o2 


255o 


2599 


2647 


2606 


2744 


48 


897 


2792 


2841 


2889 
3373 


2938 


2986 


3o34 


3o83 


3i3i 


3ioo 


3228 


48 


898 


3276 


3325 


3421 


3470 


35i8 


3566 


36i5 


3663 


37U 


48 


899 


3760 


38o8 


3856 


3905 


3953 


4001 


4049 


4098 


4146 


4194 


48 


900 


954243 


4291 


4339 


4387 


4435 


4484 


4532 


458o 


4628 


4677 

5i58 


48 


901 


4725 


4773 


4821 


4869 


4918 


4966 


5oi4 


5o62 


5iio 


48 


902 


5207 


5255 


53o3 


535i 


5399 


5447 


5495 


5543 


5592 


5640 


48 


903 


5688 


5736 


5784 


5832 


588o 


5928 


5976 


6024 


6072 
6553 


6120 


48 


904 


6168 


6216 


6265 


63i3 


636i 


6409 


6457 


65o5 


6601 


48 


905 


6649 


6697 

7176 


6745 


6793 


6840 


6888 


6936 


6984 


7082 


7080 


48 


906 


7128 


7224 


7272 


7320 


7368 


7416 


7464 


7012 


7559 


48 


907 


8086 


7655 


7703 


775i 


7799 


7847 


7894 


7942 
8421 


^468 


8o38 


48 


908 


8i34 


8181 


8229 


8277 


8325 


8373 


85i6 


48 


909 


8564 


8612 


8659 


8707 


8755 


88o3 


885o 


8898 


8946 


8994 


48 


910 


959041 


9089 


9137 


9185 


9232 


9280 


9328 


9375 


9423 


9471 


48 


911 


9518 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 


48 


912 


,9995 


••42 ••go 


•i38 


*i85 


•233 


•280 


•328 


•376 


•423 


48 


913 


960471 


o5i8 


o566 


o6i3 


0661 


0709 


0756 


0804 


o85i 


0899 


48 


914 


0946 


0994 


1 041 


1089 


ii36 


1184 


I23l 


1279 
1753 


i326 


1374 


47 


915 


1421 


1469 


i5i6 


1 563 


1611 


1 658 


1706 


1801 


1848 


47 


916 


1895 


1943 


1990 


2o38 


2o85 


2l32 


2180 


2227 


2275 


282-2 


47 


9n 


2369 
284i 


2417 


2464 


25ll 


2559 


2606 


2653 


2701 


2748 


2795 


47 


918 


2890 


2937 


2985 


3o32 


3079 


3i26 


3174 


3221 


3268 


47 


919 


33i6 


3363 


3410 


3457 


35o4 


3552 


3599 


3646 


3693 


3741 


47 


920 


963788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


4i65 


4212 


47 


921 


4360 


4307 


4354 


4401 


4448 


4495 


4542 


4590 


4637 


4684 


47 


022 


4731 


4778 


4825 


4872 


4919 


4966 


5oi3 


5o6i 


5io8 


5i55 


47 


923 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


553 1 


5578 


5625 


47 


924 


5672 


5719 


5766 


58i3 


d86o 


5907 


5954 


6001 


6048 


6095 


47 


925 


6142 


6l8q 


6236 


6283 


6329 


6376 


6423 


6470 


65i7 


6564 


47 


926 


661I 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 


47 


til 


7080 


7127 


7173 


7220 


7267 


73 14 


736i 


7408 


7454 


7501 


47 


7548 7595 

8oi6| 8062 


7642 
8109 


7688 


7735 
8203 


7782 
8249 


7829 
8296 


7875 
8343 


7922 


795? 


47 


9?9 


81 56 


8390 


8436 


47 


980 


968483 853o 


8576 


8623 


8670 


8716 


8763 


8810 


8856 


8903 


47 


931 


89501 8996 


9043 1 909c 1 


9i36 


91 83 


9229 


9276 


9323 


9869 


47 


932 ; 


94161 9463 i 


gSoc 


9556 


9602 


9649 


9695 


9742 


9789 9835 1 


47 


934 


9882 
Q70347 


9928 
0393 


9975 
0440 


••21 

0486 


••68 
o533 


•ii4 
0579 


•161 
0626 


•207 
0672 


•254 


•3oo 
0765 


ii 


o35 : 


0812 


o858 0904 1 


0951 


0997 


1044 


1090 


1137 


ii83 


1229 


46 


936 


1276 i322 1 i369 1 


i4i5 


1461 


i5o8 


1 554 


1601 


1647 


1693 


46 


937 


1740 


1786 1 i832 


1879 


1925 


1971 


2018 


2064 


2110 


2x57 


46 


93^ 


2203 


2249 22q5 


2342 


2388 


2434 


2481 


2527 


2573 


2619 


46 


939 

N. 


2666 


2712 , 2758 


2804 


285 1 


2897 


2943 


2989 


3o35 


3082 


46 





I 


2 1 


3 


4 


5 


6 


7 


8 


9 


D. 



26 



16 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 


1 I 


2 


3 


4 


5 


6 


7 


8 


9 


"46 


940 


973128 


3174 


3220 


3266 


33i3 


3359 


34o5 


345 1 


3497 


3543 


941 


35qo 
4o5i 


3636 


3682 


3728 


%ii 


3820 


3866 


3oi3 


3959 


4oo5 


46 


94a 


45?? 


4143 


4189 


4281 


4327 


4374 1 4420 


4466 


46 


943 


45i2 


4604 


465o 


4696 

5i56 


4742 


4788 ; 4834 ' 4880 


4926 
5386 


46 


944 


4972 
5432 


5oi8 


5o64 


5iio 


5202 


5248 


5294 


5340 


46 


945 


5478 


5524 


5570 


56i6 


5662 


5707 


5753 


5790 


5845 


46 


946 


5891 


6396 
6854 


5983 


6029 


6075 


6121 


6167 
6625 


6212 


6258 


63o4 


46 


947 
948 


635o 


6442 


6488 


6533 , 6579 


6671 


6717 


6763 


! 46 


6808 


6900 
7358 


6946 


6992 i 7037 


7083 


7129 


7175 


7220 


46 


949 


7266 


7312 


74o3 


7449 


7495 


7541 


7586 


7632 


7678 


46 


95o 


977724 
8181 


7769 


7815 
8272 


7861 


7906 


7952 
8409 


7998 


8043 


8089 


8i35 


46 


95 1 


8226 


83i7 ! 8363 


8454 


85oo 


8546 


8591 


46 


962 


8637 


8683 


8728 


8774 > 8819 


8865 


891 1 


8956 


9002 


9047 


46 


953 


9093 


9i38 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


95o3 


46 


954 


9548 


9594 


9639 


9685 


9730 


9776 


9821 


9867 


K^ 


9958 


46 


955 


980003 


0049 
o5o3 


0094 


0140 


oi85 


023l 


0276 


0322 


0412 


45 


956 


0458 


o549 


0594 


0640 


o685 


0730 


0776 


0821 


0867 


45 


957 
958 


0912 
i366 


0957 


ioo3 


1048 


1093 


1 139 


1 184 


1229 

1 683 


1275 


l320 


45 


141 1 


1456 


i5oi 


1 547 


1592 


1637 


1728 


1773 


45 


959 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2i35 


2181 


2226 


45 


960 


982271 


23i6 


2362 


2407 


2452 


2497 


2543 


2588 


2633 


2678 


45 


961 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3 040 


3o85 


3i3o 


45 


962 


3175 


3220 


3265 


33io 


3356 


3401 


3446 


3491 


3536 


358i 


45 


963 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3q42 
4392 


3987 


4o32 


45 


964 


4077 


4122 


4167 


4212 


4257 


43o2 


4347 


U^l 


4482 


45 


965 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


mi 


4932 
5382 


45 


966 


4977 


5022 


5067 


5ll2 


5i57 


5202 


5247 


5292 


5337 


45 


967 
968 


5426 


5471 


55i6 


556i 


56o6 


565 1 


5696 


5741 


5786 


5830 


45 


5875 


5920 

6369 


5965 


6010 


6o55 


6100 


6144 


6189 


6234 


6279 


45 


969 


6324 


641 3 


6458 


65o3 


6548 1 6593 


6637 


6682 


6727 


45 


970 


986772 


6817 


6861 


6906 


6951 


6996 


7040 


7085 


7i3o 


7175 


45 


971 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 
8024 


7622 
8068 


45 


972 


7666 
8ii3 


7711 
8i57 


2756 
8202 


7800 


7845 
8291 


7890 


838 1 


797? 
8425 


45 


973 


8247 


8336 


8470 


85i4 


45 


974 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


^S7^ 


8qi6 
936i 


8960 


45 


975 


9000 


9049 


9094 


9i38 1 9i83 


9227 


9272 


93i6 


94o5 


45 


976 


945o 


9494 


9539 


9583 9628 


9672 


9717 


9761 


9806 


9850 


44 


977 
978 


9895 


99h 
0383 


9983 


••28 


••72 


•117 


•161 


•206 


•25o 


•294 


44 


9903^9 


0428 


0472 


o5i6 


o56i o6o5 


o65o 


0694 
1137 


0738 


44 


979 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1182 


44 


980 


991226 


1270 


i3i5 


1359 


i4o3 


1448 


1492 


1 536 


i58o 


1625 


44 


981 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 


44 


983 


2111 


2i56 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


25o9 


44 


983 


2554 


25q8 
3oJ9 


2642 


2686 


2730 


2774 


2819 


2863 


2?°I 


2951 
3392 
3833 


44 


984 


2995 


3o83 


3127 


3172 


32i6 


3260 


33o4 


3348 


44 


985 


3436 


3480 


3524 


3568 


36i3 


3657 


3701 


3745 


3789 


44 


986 


3877 


3921 
436i 


3965 


4009 


4o53 


4097 


4141 


4i85 


4229 


4273 


44 


! 981 


4317 


44o5 


4449 


4493 


4537 


458i 


4625 


4669 


47i3 


a 


988 


4757 


4801 


4845 


4889 


4q33 


4977 


5o2I 


5o65 


5io8 


5i5» 


44 


989 


5196 


5240 


5284 


5328 


5372 


5416 


5460 


55o4 


5547 


5591 


44 


990 


995635 5679 


5723 


5767 


58ii 


5854 


5898 


5942 


5986 


6o3o 


44 


991 


6074 


6117 


6i6i 


62o5 


6249 


6293 


6337 


638o 


6424 6468 


44 


992 


65i2 


6555 


6599 


6643 6687 


6731 


6774 


6818 


6862 


6906 


44 


993 


6949 


6993 


7037 


7080 7124 


7168 


7212 


7255 


7209 


7343 


44 


994 


7386 


7430 


7474 


7517 7561 


7605 


7648 
8o85 


7692 


7736 
8172 


ll]l 


44 


995 


7823 


83o3 


8347 


7954 
8390 


7908 
8434 


8041 


8129 


44 


996 


8259 


8477 


8521 


8564 


8608 


8653 


44 


997 
998 


86q5 


8739 


8782 


8826 


8869 


8913 


8o56 
9392 


9000 


9043 


9087 


44 


91J1 


9174 


9218 


9261 


93o5 


9348 


9435 


9479 
991^ 


9522 


44 


999 


9565 


9609 


9652 


9696 


9739 


9783 


9826 


9870 


9957 


43 
1). 


N. 





I 


3 


3 


4 


5 


6 


7 


8 


9 



A TABLE 

OP 

LOGAKITHMIC 
SINES AND TANGENTS 

FOB EVZBl 

DEGREE AND MINUTE 
OF THE QUADEANT. 



Kemark. The minTites in the left-hand column of eaes 
page, increasing downwards, belong to the degrees at the 
top ; and those increasing upwards, in the right-hand column, 
belong to the degrees below. 



18 



^0 DEGREES.; A TABLE OF LOGARITHMIC 



M. 


Sine 


I D. 


1 Cosine 

1 


D. 


Tang. 


D. 


1 CotAHg. 











lO-OOOOOO 




i 0- 000000 




i Infinite. 


60 


I 


6-403726 


5017-17 
' 2934-85 


000000 


-00 


i 6-468726 


1 5017-17 


1:8-586274 5o 
285244 58 , 


3 


764756 


000000 


-00 


! 764756 


2984-88 


3 


940847 


2082-81 


000000 


-00 


940847 


2082-81 


059153 5-7 
12-984214 56 


4 


7-065786 


i6i5 IT 


000000 


• 00 


7-065786 


i6i5-i7 


5 


162696 


i3i9-68 


000000 


-00 


162696 


i3i9.69 


837804 j 55 1 
758122 ' 54 1 


6 


241877 


iii5-75 


9-999999 


•01 


! 241878 


1110-78 


2 


308824 


: 066 -58 


999999 


-01 


i 308825 


096-53 
852-54 


691175 1 53 i 

; 688188 ; 52 ' 


3668iG 


1 852-54 


999999 


01 


1 866817 


9 


. 417968 


762-68 


, 999999 
999998 


.01 


417970 


762-68 


58208c , 5i i 


10 


468725 


689-88 


•01 


468727 


689-88 


586278 


5o 


II 


7-5o5ii8 


629-81 


9.999998 


.01 


7 '5051 20 


629-81 


li ■494880 


^9 

48 


IS 


542906 


579-86 


999997 


.01 


542909 


579-88 


407091 


i3 


577668 


1 586-41 


999997 


.01 


577672 


536-42 


422828 


1 47 


14 


609853 


499-38 


999996 


.01 


609857 


499-89 


890148 


, 46 


i5 


689816 


• 467-14 


1 999996 


.01 


689820 


467-15 


860180 


45 


i6 


667845 


488-81 


999995 


.01 


667849 


488-82 


832i5i 


U 


;j 


694173 


418-72 


999995 


• 01 


694179 


4i3^73 


8o582i 


43 


718997 


891-35 


999994 


•01 


719004 


891-36 


280997 


42 


19 


742477 


871-27 
353.15 


999998 


-01 


742484 


371-28 


257O16 


41 


20 


764754 


999998 


-01 


764761 


35i^36 


235289 


40 


21 


7-785943 
806146 


836-72 


9.999992 


•01 


7-785951 
806 1 55 


386 • 73 


I ?• 214049 


89 


' 22 


821.75 


999991 


.01 


821-76 


198840 


38 


23 


825451 


808 -o3 


999970 
9999^ 
999988 


-01 


825460 


3o8-o6 


174540 


36 1 


24 


848984 


2o5-47 
383-88 


•02 


848944 


295-49 


i56o56 


25 


861662 


•02 


861674 


288-90 


188826 


35 


36 


878695 
895085 


278-17 
263-28 


999988 


-02 


878708 


278-18 


121292 


34 


u 


999987 
999986 


-02 


895099 


263-25 


104901 


33 


910879 


258-09 


•02 


910894 
9261^4 


254-01 


089106 


82 


29 


926119 


245-38 


999985 


-02 


245-40 


078866 


3i 


3o 


940842 


287-33 


999983 


-02 


940858 


287-35 


059142 


3o 


3i 


7-955082 


229-80 


9-999982 


-02 


7^955100 


229-81 


12*044900 


20 
28 


32 


968870 


222-78 


999981 


•02 


968889 


222-75 


081111 


33 


982233 


216-08 


999980 


.02 


982253 


216-10 


017747 


27 


34 


995198 


209-81 


999979 


-02 


995219 


200-88 
208-92 


004781 


26 


35 


8-007787 


203-90 
198.31 


999977 
999976 


-02 


8-007809 


11-992191 


25 


36 


020021 


-02 


020045 


198-88 


979955 


24 


ll 


081Q19 
043DOI 


198-02 


999975 


•02 


081945 


198-05 


968055 


23 


188-01 


999978 


-02 


048027 


188-08 


956473 


22 


3o 


054781 


188.25 


999972 


-02 


054809 


188-27 


945191 


31 


40 


065776 


178.72 


999971 


.02 


o658o6 


178-74 


934194 


20 


41 


8 -07650c 


174-41 


9-999969 


-02 


8-076531 


174-44 


11-928460 
9i8ooi 


\l 


42 


086965 


170-81 


999968 


-02 


086997 


170-84 


43 


097183 


166-89 


999966 


-02 


097217 


166-42 


902783 
892707 


\l 


44 


107167 
I 16926 


162-65 


999964 


•o3 


107202 


162-68 


45 , 


:59-o8 


999968 


•o3 


116968 


159-10 


888087 


i5 


46' 


1 2647 1 


155..66 


999961 


•o3 


i265io 


155-68 


878490 


14 


47 i 


i358io 


157.3S 


999950 


•08 


i3585i 


i52-4i 


864149 


i3 


48 ; 


144953 


149-24 


999958 


•o3 


144996 


149^27 


855oo4 


12 


49 


153907 


146-22 


999956 


•o3 


158952 


146-27 


846048 


II 


56 


162681 


143.33 


999954 


•o3 


162-727 


143-36 


887278 


10 


5i 


3.171280 


140-54 


9-999952 


•o3 


8-171828 


140-57 


11-828672 


t 


52 1 


179713 


187-86 
135.29 


999950 


• o3 


170768 j 
188086 


187-90 
135-32 


820287 


53 


187985 


999948 


•o3 


811964 
808844 


I 


54 


196102 


182.80 


999946 


• 03 


196156 ' 


182-84 


55 


204070 


180-41 


999944 


•o3 


204126 


180-44 


795874 


5 


56 


2x1895 ) 
219581 


128-10 


999942 


•04 


211953 


128-14 \ 


788047 


4 


57 


125.87 


999940 


•04 


219641 


125-90 i 


-780859 
772800 


3 


58 


227184 


128.72 


999988 


• 04 


22-7195 i 


128-76 


3 


59 


234557 
24i855 


121-64 


999986 


•04 


284621 1 


121-68 


-765379 


I 


66 


119-68 


999984 


• 04 


241921 [ 


119-67 


■758079 







Cosine 


D. 


Sine 1 




Cotang. 


D. i 


Tang. 


M. 



(89 DEORSEB.) 







SINES AND TANGENTS. (1 DEGREE.^ 




u 





8iDe 


D 




Cosino 


D. 


Tang. 


D 




Cotnng. 




8-241855 


119 


-63 


9.999934 


-04 


8-241921 


119 


-67 


11-758079 


60 


I 


249033 


"I 
ii5 


• 68 


999932 


-04 


249102 


117 


-72 
-84 


750898 


U 


a 


256094 


.80 


999929 


-04 


256i65 


ii5 


743835 


3 


263042 


ii3 


.98 


999927 


-04 


263ii5 


114 


-02 


736885 


57 


4 


269881 


112 


•21 


999925 


-04 


269956 


112 


-25 


i 730044 j 56 


5 


276614 


110 


• 50 


999922 


-04 


276691 


110 


•54 


1 723309 55 


6 


283243 


108 


• 83 


999920 


-04 


283323 


108 


•87 


716677 54 


I 


289773 


107 


• 21 


999918 


-04 


289856 


107 
io5 


-26 


710144 1 53 


296207 


io5 


65 


999915 


-04 


296292 


-70 


703708 


52 


<; 


302546 


104 


i3 


999913 


04 


302634 


104 


-18 


697366 


5i 


10 


308794 


102 


66 


999910 


04 


308884 


102 


-70 


691 I 16 


5o 


II 


8.314904 


lOI 


22 


9.999907 
999905 


04 


8-3i5o46 


101 


.26 


11-684954 


^9 


12 


321027 


99 


82 


04 


32II22 


99 


.87 


678878 
672886 


48 


i3 


327016 


98 


47 


999002 


04 


327II4 


98 


.5i 


47 


14 


332924 


95 


14 


999899 


o5 


333o25 


97 


.19 


666975 


46 


i5 


338753 


86 


999897 


o5 


338856 


95 


90 


661 144 


45 


i6 


344504 


94 


60 


999894 


o5 


344610 


94 


65 


655390 


44 


\l 


35oi8i 


93 


38 


999801 


o5 


350289 


93 


43 


6497 1 1 


43 


355783 


92 


i? 


999888 


o5 


355895 


92 


24 


6441 o5 


42 


19 


36i3i5 


91 


999885 


o5 


36i43o 


91 


08 


638570 


41 


ao 


366777 


89 


90 


999882 


o5 


366895 


89 


95 


633 I o5 


40 


ai 


8.372171 


88 


80 


9.999879 


o5 


8-372292 


88 


85 


11-627708 


39 


aa 


377499 


ll 


72 


999876 


o5 


377622 


87 


77 


622378 


38 


23 


382762 


67 


999873 


o5 


382889 


86 


72 


617111 


ll 


24 


387962 


85 


64 


999870 


o5 


3880Q2 


85 


70 


61 1908 


25 


393101 


84 


64 


999867 


o5 


393234 


84 


70 


606766 


35 


36 


398179 


83 


66 


999864 


o5 


398315 


83 


71 


601685 


34 


11 


4o3i99 


82 


71 


999861 


o5 


4o3338 


82 


t 


596662 


33 


408161 


81 


ll 


999858 


o5 


4o83o4 


81 


591696 


32 


29 


4i3o68 


80 


999854 


o5 


4i32i3 


80 


91 


586787 


3i 


3o 


417919 


79 


96 


999851 


06 


418068 


80 


02 


581932 


3o 


3i 


8-422717 


72 


09 


9-999848 


06 


8-422860 


79 


14 


ii-577i3i 


29 


32 


427462 


78 


23 


999844 


06 


427618 


78. 


3o 


572382 


28 


33 


432 1 56 


]l 


40 


999841 


06 


4323i5 


77- 


45 


567685 


ll 


34 


436800 


57 


999838 


06 


436962 


76 


63 


563o38 


35 


441394 


75 


77 


999834 


06 


441 56o 


75 


83 


558440 


25 


36 


445941 


74 


99 


999831 


06 


446110 


75. 


o5 


5538Q0 


24 


3? 


450440 


74 


22 


999827 


06 


45o6i3 


74- 


28 


549387 


23 


36 


454893 


73 


46 


999823 


06 


455070 


73- 


52 


544930 
540619 


aa 


39 


459301 
463665 


72 


73 


999820 


06 


459481 
463849 


72- 


79 


21 


40 


72 


00 


999816 


06 


72- 


06 


536i5i 


20 


41 


8-467985 


71 


29 


9-999812 • 


06 


8-468172 


71- 


35 


11-531828 


;? 


42 


472263 


70 


60 


999809 


06 


472454 


70- 


66 


52-7546 


43 


476498 


69 


91 


999803 


06 


476693 


69- 


98 


523307 


\i 


44 


480693 


69 


24 


999801 


06 


480892 


69- 


3i 


519108 


45 


484848 


68 


59 


999797 • 


07 


485o5o 


68- 


65 


5i4o5o 


i5 


46 


488963 


67 


94 


999793 • 


07 


489170 


68- 


01 


5io83o 


14 


s 


493040 


67 


3i 


999700 - 


07 


493250 


67- 


38 


506750 


i3 


497078 
5oio8o 


66 


60 


999786 ■ 


07 


497293 


66- 


ii 


502707 


12 


49 


66 


08 


999782 - 


07 


501298 


66- 


i5 


498702 


II 


5o 


5o5o45 


65 


48 


999778 . 


07 


505267 


65- 


55 


494733 


10 


5i 


8 -508074 


64 


89 


9.999774 - 


07 


8-50Q200 


64- 


39 


11-490800 


§ 


52 


512067 


64 


3i 


999769 - 


07 


51^098 


64- 


486902 


53 


516726 


63 


75 


999765 - 


07 


516961 


63- 


82 


483o39 


I 


54 


52o55r 


63 


19 


999761 - 


07 


520790 


63- 


26 


479210 
475414 


55 


524343 


62 


64 


999757 • 
999753 - 


07 


524586 


62- 


72 


5 


56 


528102 


62 


II 


07 


528349 


62. 


18 


47i65i 


4 


5^ 


531828 


61 


58 


999748 - 


07 


532080 


61- 


65 


467920 


3 


535523 


61 


06 


999744 


07 


535779 


61- 


i3 


464221 


2 


59 


539186 


60 


55 


999740 • 


07 


539447 
543084 


60- 


62 


460553 


1 


60 


542819 


60 -04 


999735 


07 


60-12 


456916 





Cosine 


D. 


Sind 1 




Cotang. 


D. 


Tang 






1 


6 




(83 D 


SOB 


J:B8.) 











20 


(2 


DEGREES.) A TABLE OF LOGARITHMIC 




M. 


Bino 


D. 


Cosino 


D. 


Tang. 


D. 


OotAPg. 


1 

_ 1 


o 


8.542819 


60.04 


9.999735 


-07 


8.548084 


60-12 


11 -456916 1 60 


I 


546422 


59.55 


999781 


.07 


546691 


59 


•62 


453309 , 59 


a 


549995 
553539 


59-06 


999726 


.07 


550268 


59 


•14 


449782 


58 


3 


58-58 


999722 


.08 


558817 


58 


•66 


446188 


57 


4 


557054 


58.11 


999717 


.08 


557886 


58 


.19 


442664 


56 


5 


56o54o 


57.65 


999718 


.08 


560828 


57 


•73 


439172 


55 


6 


563999 


57.19 


999708 


.08 


564291 


57 


•27 


4357OQ 
482278 


54 


I 


567431 


56-74 


999704 


.08 


567727 


56 


.82 


53 


570886 


56.80 


999699 


.08 


571187 


56 


•38 


428863 


52 


9 


574214 


55.87 


999604 


.08 


574520 


55 


•95 


425480 


5i 


10 


577566 


55.44 


999689 


.08 


577877 


55 


•52 


422123 


5o 


II 


8.580892 


55.02 


9.999685 


.08 


8-581208 


55 


10 


II -418702 


49 


12 


584193 


54.60 


999680 


-08 


5845i4 


54 


68 


415486 


48 


i3 


587469 


54.19 


999675 


-08 


587795 


54 


27 


4l2205 


ii 


14 


590721 


53.79 
53.39 


999670 


-08 


591031 


53 


.87 


408949 


i5 


598948 


999665 


-08 


594288 


53 


47 


405717 
402 5o8 


45 


i6 


597152 


53.00 


999660 


-08 


597492 


53 


08 


44 


\l 


600882 


52.61 


999655 


-08 


600677 


52 


70 


899328 


43 


608489 
606628 


52-28 


999650 


-08 


608880 


52 


32 


896161 


42 


19 


51.86 


999645 


-09 


606978 


5i 


94 


893022 


41 


20 


609784 


5i 49 


999640 


.09 


610094 


5i 


58 


389906 


40 


31 


3.612828 


51-12 


9.999635 


-09 


8-618189 


5i 


21 


11-386811 


89 


32 


615891 
618987 


50.76 


999629 


-09 


616262 


5o 


85 


888788 


38 


33 


50.41 


999624 


-09 


619818 


5o 


5o 


880687 


3? 


24 


621962 


5o.o6 


999619 


-09 


622843 


5o 


i5 


377657 
874648 


25 


624965 


49-72 


999614 


-09 


625352 


49 


81 


35 


26 


627948 


49.88 


999608 


-09 


628840 


49 


47 


871660 


34 


11 


680911 
633854 


49-04 


999608 


-09 


681808 


49 


i3 


368692 


33 1 


48.71 


999597 


-09 


634256 


48 


80 


365744 
862816 


32 1 


?9 


686776 
689600 


48-89 


999592 


.09 


687184 


48 


48 


3i 


3o 


48.06 


999586 


.09 


640098 


48 


16 


359907 


3o 


3i 


8.642563 


47-75 


9.999581 


.09 


8-642982 


47 


84 


11-357018 


It 


32 


645428 


47-43 


999575 


-09 


645858 


47 


53 


354147 
351296 


33 


648274 


47-12 


999570 


-09 


648704 


47 


22 


11 


34 


65iio2 


46-82 


999564 


.09 


65i587 


46 


91 


848468 


35 


6539 II 


46-52 


999558 


'10 


654852 


46 


61 


345648 


25 


36 


656702 


46-22 


999553 


•10 


657149 


46 


3i 


342851 


24 


^7 
3S 


659475 
662230 


45-92 


999547 


-10 


659928 


46 


02 


340072 


23 


45-63 


999541 


-10 


662689 


45 


73 


387811 


22 


39 


664968 


45-35 


999535 


• 10 


665483 


45 


44 


384567 


21 


40 


667689 


45.06 


999529 


• 10 


668160 


45 


26 


881840 


20 


41 


8.670398 


44.79 


9.999524 


-10 


8-670870 


44 


88 


11-829180 


19 


42 


678080 


44-51 


999518 


-10 


673563 


44 


61 


826487 


18 


43 


675751 


44-24 


999512 


-10 


676289 


44 


34 


828761 


\l 


44 


678405 
681043 


43-97 


999506 


.10 


678900 


44 


17 


821100 


45 


43.70 


999500 


.10 


681544 


43 


80 


3 I 8456 


i5 


46 


688665 


43.44 


999498 


.10 


684172 
686784 


43 


54 


; i5828 


14 


I 47 
; 48 


686272 


43.18 


999487 


.10 


43 


28 


ii32i6 i3 j 


688868 


42.92 


999481 


•10 


689881 


43 


o3 


810619 


13 


49 


691488 


42.67 


999475 


.10 


691968 


42 


77 


808087 


II 


5o 


698998 


42.42 


999469 


-10 


694529 


42 


52 


805471 


ro 


5i 


8-696548 


42-17 


9.999463 


-11 


8-697081 


42 


28 


11 802919 


9 ! 


5i 


699078 
701589 


41-92 


999456 


.11 


699617 


42- 


o3 


3oo383 8 


53 


41.68 


999450 


.11 


702189 


41- 


It 


297861 1 
295354 


54 


704090 


41.44 


999443 j 


• II 


704646 


41- 


55 


706577 


41.21 


999437 


• 11 


707140 


41- 


32 


292860 5 


56 


709049 


40.97 


999481 


• II 


709618 


4i- 


08 


200882 4 


u 


71 1 507 
718952 
716383 


40-74 
40. 5i 


999424 
999418 


• II 

• II 


712088 
714534 


40- 
40- 


85 
62 


287917 
285465 


3 
3 


59 


40.29 


999411 


• II 


716072 
719896 


40- 


40 


288028 


I 


6o 


718800 


40'06 


999404 


• II 


40-17 


280604 







CoBino 


D. 


Sine 




Cotang. 


D. 


Tang. 


M.] 








(87 p 


SGR 


ESS.) 











SINES AND TANGENTS (3 DEGREES., 



21 



M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 





8.718800 


4o«o6 


9.999404 


II 


8.719896 


40-17 


1 1 - 280604 


I 


721204 


39 


84 


999898 


•II 


721806 


89-95 


278194 


^ 


3 


723595 


39 


6a 


999891 


11 


724204 


89 •74 
89.52 


275796 


3 


730688 


39 


41 


999884 


• 11 


726588 


278412 


U 


4 


89 


19 


999878 


•II 


728959 


89-80 


27 1 041 


5 


88 


98 


999871 


•11 


781817 


89-09 


268683 


55 


6 


733027 


88 


77 


999864 


•12 


783663 


38-89 


266337 


54 




735354 


88 


u 


999357 ; 


•12 


735996 


38-68 


264004 


53 


737667 


88 


999830 


'12 


70.8817 


38.48 


261688 i 5a 1 


9 


739969 


38 


16 


999843 


12 


740626 


38-27 


259874 


5i 


10 


742259 


37 


96 


999886 


12 


742922 


38-07 


257078 


5o 


II 


8-744536 


37 


76 


9.999829 


12 


8.745207 


37-87 


1 1 - 254793 


it 


la 


746802 


37 


56 


999822 


•12 


747479 


87-68 


252521 


i3 


749055 


37 


37 


999815 


12 


749740 


37-49 


25o26o 


47 


14 


751297 
753528 


37 


17 
98 


999808 


•12 


751989 


87-29 


248011 


46 


i5 


36 


999801 


12 


754227 


87-10 


245778 


45 


i6 


755747 


36 


79 


999294 


12 


756453 


86-92 


243547 


44 


\l 


757955 


36 


61 


999286 


12 


758668 


86.73 


241882 


43 


76oi5i 


86 


42 


999279 


12 


760872 


36-55 


289128 


42 


19 


762337 


36 


24 


999272 


12 


768065 


36.36 


286935 


41 


20 


76451 I 


36 


06 


999265 


12 


765246 


36.18 


284754 


40 


21 


8.766675 


35 


88 


9-999257 


12 


8-767417 
769578 


36-00 


11.232588 


89 


22 


768828 


35 


70 


999250 


l3 


35.88 


280422 


88 


23 


770970 


35 


53 


999242 


18 


771727 


35-65 


228278 
226134 


^I 


24 


773101 


85 


35 


999235 


l3 


778866 


35-48 


36 


25 


775223 


85 


18 


999227 


18 


775995 


35-3i 


224oo5 


35 


26 


777333 


35 


01 


999220 


18 


778114 
780222 


35-14 


221886 


34 


11 


779434 


34 


84 


999212 


18 


34-97 


219778 


33 


78i524 


34 


67 


999205 


18 


782820 


84.80 


217680 


32 


V^ 


7836o5 


34 


5i 


999197 


•18 


784408 


34.64 


215592 


3i 


3o 


785675 


34 


3i 


999189 


18 


786486 


34-47 


2i85i4 


3o 


3i 


8.787736 


34 


18 


9-999181 


18 


8.788554 


34.31 


I I- 21 1446 


29 


32 


789187 
791828 


34 


02 


999174 


i3 


790618 


84-15 


209887 


28 


33 


33 


86 


999 I 66 


i3 


792662 


33.99 


207888 


27 


34 


793359 


33 


70 


999 1 58 


18 


794701 


38.88 


205299 


26 


35 


795881 


33 


54 


999 I 5o 


i3 


796781 


83.68 


208260 


25 


36 


797894 


88 


8q 


999 I 42 


i3 


798752 
800768 


33.52 


201248 


24 


u 


799897 
801892 


88 


26 


999184 


i3 


33.87 


199287 


23 


38 


08 


999 1 26 


i3 


^2765 


33.22 


197285 


22 


39 


808876 


82 


93 


099118 


i3 


804758 


33.07 


195242 


21 


40 


8o5852 


32 


78 


9991 10 


i3 


806742 


82.92 


198258 


20 


41 


8-807819 


32 


63 


9.999102 


i3 


8-808717 


82-78 


11-191283 


19 


42 


809777 


32 


49 


999094 


14 


8io683 


82.62 


189817 


18 


43 


811726 


32 


34 


999006 


14 


812641 


82.48 


187359 


n 


44 


818667 


82 


19 
o5 


999077 


14 


814589 


32-33 


1 8541 1 


16 j 


45 


815599 


82 


999069 


14 


816529 


32-10 


188471 


i5 ' 


46 


817522 


3i 


91 


999061 


14 


818461 


32-o5 


181539 
179616 


14 


J? 


819486 


3i 


77 


999053 


14 


820884 


81-91 


i3 


821843 


3i 


63 


999044 


14 


822298 


81-77 
3i-63 


177702 
175795 
173891 


12 


49 


828240 


3i 


ii 


999086 


14 


824205 


II 


5o 


825i3o 


3i 


999027 


14 


826108 


3i-5o 


10 


5. 


8-827011 


3i 


22 


9.999019 


14 


8-827992 


3i-36 


11-172008 


? 


52 


828884 


3i 


08 


999010 


14 


829874 


31-23 


170126 


53 


880749 


80 


95 


999002 


14 


831748 


3i-io 


168252 


I 


54 


882607 


3o 


82 


998998 


14 


888618 


80-96 


166887 


55 


884456 


3o 


69 


998984 


14 


885471 


30-83 


164529 


5 


56 


886297 
888180 


3o 


56 


998976 


14 


887821 


30-70 
30-5-7 
3o-45 


J162679 
160887 


4 


S 


3o 


43 


998967 


i5 


889168 


3 


889956 


3o 


80 


998958 


i5 


^840998 
842825 


159002 


a 


^ 


841774 
843585 


3o 


17 


998950 


i5 


3o.3a 


157175 


I 


60 


3o-oo 


998941 


i5 


844644 


30-19 


155356 





Cosine 


D. 


Sine 




Cotang. 1 


D. 


Tang. 


M. 



(86 DEGRESS.) 



22 


(4 


DEGREES.) A 


TABLE OF LOaARITHMIC 




M. 




Sine 


D. 


Coeino 


D. 


Tang. 


B. 


Colang. 


bo 


8.843585 


3o.o5 


9-998941 


• 15 


8-844644 


80.19 


11- 155356 


I 


845387 
847183 
848971 
85o75i 
8525a5 


29.92 


998982 


•i5 


846455 


80.07 


153545 5o 


2 


29.80 


998923 


.i5 


848260 


1 29.05 


i5i74o 56 


3 


29-67 


998914 


.15 


85oo57 


29.82 


149943 
1 148154 


U 


4 


29.55 


998905 


.15 


85 1 846 


29.70 


1 5 


29.43 


998896 


-15 


853628 


29-58 


146872 


55 


6 


854291 


29.81 


998887 


.15 


855403 


29.46 


144597 


54 


I 


856049 


29.19 


998878 


.15 


857171 


29-35 


142829 


53 


857801 


29.07 


998869 


.15 


858982 


29-28 


141068 


52 


9 


859546 


28.96 


998860 


• 15 


860686 


29-11 


189814 


5i 


10 


861283 


28.84 


998851 


-15 


862488 


29-00 


187567 


5o 


II 


8-863oi4 


28.78 


9-998841 


.i5 


8-864178 


28-88 


11-135827 


t 


12 


864738 


28.61 


098882 


.i5 


865906 


28-77 


184094 


i3 


866455 


28.50 


998828 


.16 


867682 


28-66 


182868 


47 


14 


868 I 65 


28.80 


998818 


.16 


869351 


28.54 


! 180649 


46 


i5 


869868 


28-28 


998804 


.16 


871064 


28.48 


\ 128986 


45 


i6 


871565 


28-17 


998705 


■16 


872770 


28.82 


, 127280 


44 


\l 


873255 


28.06 


998785 


.16 


874469 


28.21 


I2553i 


43 


874938 


27.05 


998776 


.16 


876162 


28.11 


128888 


42 


19 


876615 


27.86 


998766 


.16 


877849 


28.00 


122l5l 


41 


20 


878285 


27.78 


998757 


.16 


879529 


27.89 


1:0471 


40 


21 


8.879949 
881607 
883258 


27.68 


9.998747 


.16 


8-881202 


mt 


11- I 18708 
117181 
110470 


39 
38 


22 


27.52 


998788 


.16 


882869 


23 


27.42 


998728 


.16 


884580 


27.58 


U 


24 


884903 


27.81 


998718 


-16 


886 I 85 


27.47 


ii88i5 


25 


886542 


27.21 


998708 


-16 


887888 


27.87 


112167 


35 


26 


888174 


27.11 


998699 


.16 


889476 


27.27 


iio524 


34 


11 


889801 


27.00 


998609 


-16 


891 1 12 


27.17 


108888 


33 


891421 


26.90 


998679 


•16 


892742 


27.07 


107258 


32 


29 


898035 


26.80 


998669 


.17 


894866 


26.97 


io5684 


3i 


3o 


894643 


26.70 


998659 


•17 


895984 


26.87 


104016 


3o 


3i 


8-896246 


-26.60 


9-998649 


•17 


8-897596 


26.77 


11.102404 


20 
28 


32 


807842 


26.51 


998689 


.17 


899208 


26-67 


100797 


33 


899432 


26.41 


998629 


•17 


900808 


26-58 


099197 


U 


34 


901017 


26.31 


998619 


.17 


902898 


26.48 


097602 


35 


902596 


26.22 


998609 


•n 


908987 


26-38 


096018 


25 


36 


904169 


26.12 


998599 


•17 


905670 


26-29 


094480 


24 


^1 


905786 


26-08 


998589 


•17 


907147 


26-20 


092853 


23 


38 


907297 


25.98 


998578 


•17 


908719 


26-10 


091 281 


22 


39 


908853 


25.84 


998568 


•17 


910285 


26-01 


089715 


21 


40 


910404 


25.75 


998558 


.17 


91 1846 


25-92 


088 1 54 


20 


41 


8.911940 


25.66 


9.998548 


•n 


8-918401 


25-88 


T» .086599 


19 
18 


42 


913488 


25-56 


998537 


.17 


914951 


25-74 


o85o49 


43 


9l5022 


25.47 


998527 


.17 


916495 


25-65 


o835o5 


\l 


44 


9i655o 


25.38 ' 


998516 


.18 


918084 


25-56 


081966 


45 


918073 


25.29 


998006 


.18 


919568 


25-47 


080482 


i5 


46 


919591 


25.20 


998495 


.18 


921096 


25-88 


078904 


14 


% 


921103 


25.12 ! 


998485 


.18 


922619 


25.80 


077381 . 1-3 1 


923610 


25 o3 


998474 


.18 


924186 


25.21 


*5864 


12 


<9 


9«4ii2 


24-94 1 


998464 


• 18 


925649 


25-12 


074851 


II 


5o 


925609 


24-86 i 


998453 


.18 


927156 


25 -o3 


072844 


10 


5i 


8.927100 


24-77 


9-998442 


.18 


8-928658 


24-95 


II 071842 


? 


52 


92858T 
980068 


24-69 


998481 


.18 


980155 


24-86 


069845 


53 


24-60 1 


998421 


.18 


981647 


24-78 


068853 


I 


54 


981544 


24-52 ! 


998410 


.18 


988184 


24-70 


066866 


55 


988015 


24-43 1 


998899 


.18 


984616 


24-61 


065884 


5 


56 


984481 


24-35 i 


998888 


.18 


086098 


24-53 


068907 
062435 


4 


ll 


935942 


24-27 1 


998877 


-18 


987565 


24-45 


3 


987898 
938850 


24-19 


998866 


-18 


989082 


24-87 


060968 
059606 


3 


59 


24-11 


998855 


.18 


940494 
941902 


24-80 


I 


60 


940296 


24-o3 


998844 


.18 


24-21 


058048 





1 Cosine 


D. 1 


Sine 




Cotang. 


D. 


Tang. 


M. 








(85 


DEOR 


EES.) 











SIXES AND TANGENTS. (5 DEGRBB., 


t 


22 


M 


Bind 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. ! 





8 043 JOv*) 


24 -03 


9.998344 


.19 


8-941962 


24-21 


M- 068048 


60 


I 


23 


.04 


998333 


.19 


948404 


24- 13 


066696 


u 


2 


943174 


23 


-07 


998822 


.19 


944852 


24 -06 


066148 


3 


944606 


28 


•79 


9988 1 1 


.19 


946295 


23^97 


068705 


u 


4 


946084 


28 


•71 


99880c 


-19 


947734 


23^90 


062266 


5 


947456 


28 


-68 


998289 


.19 


949168 


23.8s 


060882 


55 


6 


Q48874 


23 


-55 


998277 


.19 


960397 


23.74 


049403 


54 ! 


7 


960287 


28 


-48 


998266 


.19 


962021 


I 23-66 


047979 


53 ; 


8 


951696 


23 


-40 


998255 


.19 


968441 


1 28-60 


046669 


52 


9 


96J100 


23 


-32 


998243 


.19 


964866 


28-5i 


046144 


5i 


10 


954499 


23 


-25 


998282 


.19 


966267 


23-44 


048733 


5o 


II 


8.955894 


23 


I? 


9 998220 


•19 


8-967674 


23-37 


11-042826 


49 


n 


957284 
958670 
960052 


23 


•10 


998209 


19 


969075 


! 23-29 

23-23 


040926 


48 


13 


23 


02 


998107 


.19 


960478 


089527 


S 


14 


22 


95 


998186 


.19 


961866 


i 23.14 


o38i34 


15 


961429 


22 


88 


998174 


.19 


968266 


23.07 


086746 


45 


16 


962801 


22 


80 


998168 


.19 


964689 


23-00 


o3636i 


44 


\l 


964170 


22 


73 


998151 


.19 


966019 


22. o3 


088981 


43 


965534 


22 


66 


998189 


• 20 


967894 


22.86 


082606 


42 


19 


966893 


22 


59 


998128 


•20 


968766 


22.79 


081284 


41 


20 


968249 


22 


02 


998116 


.20 


970188 


22.71 


029867 


40 


21 


8-969600 


22 


44 


9-998104 


•20 


8-971496 


22.65 


11-028604 


i% 


22 


970947 


22 


88 


998092 


•20 


972836 


22-67 


027145 


38 


23 


972289 


22 


3i 


998080 


•20 


974209 


22-51 


026791 


il 


24 


978628 


22 


24 


998068 


•20 


976660 


22.44 


024440 


36 


35 


974962 


22 


n 


998056 


.20 


976906 


22.37 


028094 
021762 


35 


26 


976298 


22 


10 


998044 


.20 


978248 


22-3o 


34 


11 


977619 


22 


o3 


998082 


•20 


979686 


22-23 


020414 


33 


978941 


21 


97 


998020 


.20 


980921 


22-17 


019079 


32 


29 


980259 
981578 


21 


90 


998008 


.20 


982261 


22-10 


017749 
016428 


3i 


3o 


21 


83 


997996 


•20 


988677 


22-04 


3o 


3i 


8-982888 


21 


77 


9.997985 


.20 


8-984899 


21.97 


ii-oi5ioi 


^2 


32 


984189 


21 


*?? 


997972 


.20 


980217 


21.91 


018788 


28 


33 


985491 


21 


63 


997959 


.20 


987682 


21.84 


012468 


u 


34 


9^^789 


21 


^7 


997947 


.20 


988842 


21.78 


011168 


35 


988088 


21 


5o 


997935 


•21 


990149 


21-71 


009861 


35 


36 


989374 


21 


44 


997922 


•21 


991461 


21-65 


008649 


24 


1 


990660 


21 


38 


997010 


.21 


992760 


21-58 


. 007260 


23 


991943 


21 


3i 


997897 


.21 


994046 


21-62 


006965 


22 


39 


998222 


21 


25 


997885 


.21 


996887 


21.46 


004663 


21 


4o 


994497 


21 


19 


997872 


.21 


996624 


21-40 


008876 


20 


41 


8-995768 


21 


12 


9-997860 


.21 


8.997908 


21.34 


11-002092 


It 


42 


997086 


21 


06 


997847 
997886 


•21 


999188 


21-27 


000812 


43 


998299 


21 


00 


•21 


9.000466 


21-21 


10-999635 

998262 


■2 


444 


999560 


20- 


q4 


997822 


•21 


001788 


21-l6 


45 


9'OOo8i6 


20- 


87 


997809 


•21 


008007 


21-09 


996998 


i5 


46 


002069 


20 


82 


997797 


•21 


004272 


21 -o3 


996728 


U 


47 


008818 


20- 


76 


997784 


• 21 


oo5534 


20-97 


994466 


i3 


48 


004563 


20- 


70 


997771 


•21 


006792 


20-91 


998208 


12 


49 


oo58o5 


20- 


64 


997758 


•21 


008047 
009298 


20-85 


991968 


II 


5o 


007044 


20- 


58 


997745 


•21 


20-80 


990702 


10 


5i 


9-008278 


20- 


52 


9-997782 


•21 


9-010646 


20-74 


10-989454 


I 


52 


009510 


20- 


46 


997719 


•21 


011790 
018081 


20-68 


988210 


53 


010787 


20- 


4e 


997706 


•21 


20-62 


986969 


I 


54 


01196a 


20- 


34 


997698 


.22 


014268 


20-56 


98678-2 


55 


018182 j 


20- 


29 


997680 


.22 


oi56o2 


20-5l 


984498 


5 


56 


014400 


20- 


23 


997667 


.22 


016782 


20-45 


988268 


4 


u 


oi56i3 1 


20- 


17 


997654 


.22 


017960 
019188 


20-40 


982041 


3 


016824 


20- 


12 


997641 


.22 


20-33 


980S17 


2 


59 


018081 


20- 


06 


997628 


.22 


020408 


20-28 


978380 


I 


^ 


019335 


20-00 


997614 


.22 


021620 


20-23 







Cofcine 


D. 


Sine 




Cotang. 


D. 


Tang^ 


M. 










^84 


DEGR 


SES.'^ 









u 



yfi DEGREES.) A TABLE OF LOGARITHmC 



M 




Sine 


D. 


Coeine 


D. 


Tang. 


D. 


Gotang. 


60 


9.019235 


30.00 


9.997614 


.22 


9.021620 


20.23 


10.978880 


I 


020435 


I9-q5 


997601 


• 22 


022884 


20 


•n 


977166 5o 
976966 58 


a 


021682 


19.89 


997588 


.22 


024044 


20 


.11 


3 


022825 


19.84 


997574 


.22 


02525l 


io 


06 


974749 57 
978645 1 56 


4 


024016 


19-78 


997661 


.22 


026455 


20 


•00 


5 


0252o3 


19-78 


997547 


.22 


027655 
028852 


19 


•95 


972845 55 


6 


026386 


19-67 


997534 


.28 


19 


-90 


971148 54 


I 


027567 


19-62 


997620 


•23 


080046 


19 


-85 


969964 53 


028744 


19.57 


997507 


.28 


081287 
082420 


19 


•79 


968768 
967676 


5a 


9 


029918 


i9-5i 


997493 


.28 


19 


•74 


5i 


IC 


081089 


19.47 


997480 


.28 


088609 


19 


69 


966891 


5o 


II 


9 -032257 


19-41 


9.997466 


-28 


9-034791 


19 


-64 


10.966209 


S 


12 


o3342i 


19.36 


997452 


-23 


035969 


»9 


-58 


964081 


i3 


034582 


19.80 


997489 


-23 


087144 


19 


•58 


962866 


47 


14 


035741 
086896 


19.25 


997425 


-23 


o388i6 


19 


48 


961684 


46 


i5 


19.20 


99741 1 


-28 


089485 


19 


.43 


960616 


45 


i6 


088048 


19.15 


997897 


.28 


04065 I 


19 


• 88 


960849 


44 


n 


039197 


19.10 


997888 


.28 


041818 


19 


■33 


968187 


43 


i8 


040842 


1Q.05 


997869 


.28 


042978 


19 


•28 


967027 


42 


19 


041485 


18.99 


997355 


.28 


o44i3o 


19 


23 


966870 


41 


20 


042625 


18.94 


997841 


.28 


045284 


19 


18 


964716 


40 


21 


9-043762 
044895 


18.89 


9.997827 
997318 


•24 


9.046484 


19 


18 


10.968666 


U 


22 


18.84 


•24 


047582 


19 


08 


9624x8 


23 


046026 


18.79 
18.75 


697209 


•24 


048727 


19 


08 


961278 
9501J1 


ll 


24 


047154 


997285 


•24 


049860 


18 


98 


25 


048279 


18.70 


997271 


•24 


o5ioo8 


18 


08 


948992 


35 


26 


049400 


i8.65 


997257 


-24 


o52i44 


18 


89 


947866 


34 


11 


o5o5i9 
o5i635 


18.60 


997242 


•24 


053277 


18 


84 


946728 


33 


18.55 


997228 


•24 


054407 


18 


79 


945598 


3a 


29 


052749 
053859 


18. 5o 


997214 


•24 


055585 


18 


74 


944466 


3i 


3o 


18.45 


997199 


-24 


o56659 


18 


70 


943341 


3o 


3i 


9-054966 


18.41 


9.997185 


•24 


9-057781 


18 


65 


10.942219 


29 
28 


32 


o56o7i 


18.36 


997170 


•24 


068900 


18 


69 


941 100 


33 


057172 

0582TI 

J059367 


18.31 


997156 


•24 


060016 


18 


55 


980984 


27 


34 


18.27 


997141 


•24 


061180 


18 


5i 


988870 


26 


35 


18.22 


997127 


•24 


062240 


18 


46 


987760 


25 


36 


060460 


I8.I7 

18. i3 


997112 


•24 


063348 


18 


42 


9366fj 


24 


ll 


o6i55i 


997008 


•24 


064453 


18 


37 


986647 


28 


062689 


18.08 


997088 


.25 


065556 


18 


3i 


984444 


22 


39 


068724 
064806 


18.04 


997068 


-25 


066655 


18 


28 


933345 


21 


40 


17.99 


997053 


.25 


067762 


18 


24 


982248 


20 


41 


9-065885 


17-94 


9.997089 


•25 


9.068846 


18 


19 


10-981154 


19 


42 


066962 


17.00 


997024 


.25 


069988 


18 


i5 


980062 


18 


43 


068086 


17.86 


997009 


.25 


071027 


18 


10 


928978 
927887 
926808 


»7 


44 


069107 


17-81 


996994 


.25 


072118 


i8- 


06 


16 


45 


070176 


n-ii 


996979 


.25 


078197 


18. 


05 


i5 


46 


071242 


17.72 


996964 


•25 


074278 


17 


P 


926722 


14 


% 


072806 


17.68 


996949 


•25 


1)75356 


17 


924644 


i3 


078866 


17.68 


^96934 


•25 


076482 


n- 


b 


928668 


12 


49 


074424 


17.59 


996919 


.25 


o775o5 
078576 


n- 


84 


922496 


11 


5o 


075480 


^7.55 


996904 


•25 


17. 


80 


921424 


13 

1 


5i 


9.076533 


17.50 


9.996889 


.25 


9.079644 


n- 


76 


10 '920356 


§ 


52 


077588 
078681 


17-46 


996874 


.25 


080710 


n- 


72 


919290 


53 


17-42 


996858 


.25 


081778 
082888 


n- 


63 


918227 


I 


54 


079676 
080719 


17-88 


996848 


.25 


n- 


917167 


55 


17-38 


996828 


.25 


088891 


17- 


59 


916100 
916063 


5 


56 


081759 


17.29 
17.25 


996812 


.26 


084947 


17 


55 


4 


U 


088882 


996707 


.26 


086000 


17 


5i 


914000 


3 


17.21 


996782 


.26 


087050 


17 


tl 


912960 


a 


59 


084864 


17.17 
17 i3 


996766 


.26 


088098 


n- 


911002 
910066 


I 


66 


085894 


996751 


.26 


089144 


17- 


38 







Coeine 


__D. 


Sine 




Cotan^. 


~'d. 


Tanfir. 



(83 DSORBES.) 





1 


SINES AND TANGENTS. (7 DEGREES.) 


2 





Sine 


D. 


CoBuie 


D. 


Tang. 


D. 


Cotang. 




9-085894 


i7-i3 


9-996751 


.26 


Q 089144 


17.38 


10-910856 


60 


I 

J 


086922 

087947 

1 088970 


17.09 
17-04 


996735 
996720 


• 26 
-26 


090187 
O9122S 


17-34 
17-80 


909813 
908772 


u 


3 


I7-00 
16-96 


996704 


-26 


092266 


17.27 


907784 
90669b 


u 


4 


089990 


996688 


• 26 


098802 


17.22 


5 


091008 


16.92 


996678 


-26 


094886 


1719 

17.15 


9o5664 


55 


6 


1 092024 


1 16.88 


996657 


-26 


095867 


904688 


54 


I 


098087 


1 16.84 


996641 


-26 


096895 


17. 11 


908605 


53 


094047 


16.80 


996625 


.26 


097422 


17.07 
17.08 


902578 
901554 


52 


Q 


095o56 


16.76 


996610 


• 26 


098446 


5i 


10 


096062 


16.73 


996594 


.26 


099468 


16.99 


900532 


5o 


II 


9-097065 


16.68 


9-996578 


-27 


9-100487 


16-95 


10-899518 


S 


12 


098066 


16-65 


996562 


.27 


ioi5o4 


16-91 


898496 


i3 


099065 


16-61 


996546 


.27 


io25i9 


16.67 


897481 


47 


14 


100062 


16-57 
16.53 


996580 


.27 


io8532 


16-84 


896468 


46 


i5 


ioio56 


996514 


.27 


104542 


16-80 


895458 


45 


i6 


102048 


16-49 


996498 


.27 


io555o 


16-76 


894450 


44 


\l 


108087 

104023 


i6-45 


996482 


.27 


io6556 


16-72 


898444 


43 


16-41 


996465 


.27 


107559 
io856o 


16-69 


892441 


42 


»9 


loSoio 


16-88 


996449 


.27 


16-65 


891440 


4i 


30 


105992 


16-84 


996488 


.27 


109559 


16-61 


890441 


40 


at 


9-106978 
107901 


i6-3o 


9-9964x7 


.27 


9-iio556 


16-58 


10-889444 


It 


22 


16-27 


996400 


•27 


iii55i 


16-54 


888449 


23 


108927 


16-23 


996884 


.27 


1 1 2543 


i6-5o 


887457 


u 


24 


I 0990 I 


16-19 


996868 


•27 


I 13588 


16-46 


886467 


25 


110873 


16-16 


996351 


.27 


I 14521 


16-43 


885479 
884493 


85 


26 


II 1842 


16-12 


996885 


.27 


11 5507 


16-89 


84 


11 


1 1 2809 


16-08 


996818 


•27 


116491 


i6-36 


888509 


33 


118774 


i6-o5 


996802 


.28 


I 17472 
118452 


16-82 


882528 


32 


29 


114787 
II 5698 


16-01 


996285 


.28 


16-29 


881548 


3i 


So 


15-97 


996269 


.28 


I 19429 


16-25 


880571 


3o 


3i 


9-ii6656 


15.94 


9-996252 


• 28 


9 - 1 20404 


16-22 


10-879596 


It 


32 


117613 


15.90 


996285 


.28 


121877 


16-18 


878623 


33 


I 18567 


15.87 


996219 


.28 


122848 


16. i5 


877652 


27 


34 


119519 


i5.88 


996202 


.28 


128817 


16-11 


876688 


26 


35 


120469 


i5.8o 


996185 


.28 


124284 


16-07 


875716 


25 


36 


121417 


15.76 


996168 


.28 


125249 


16-04 


874751 


24 


ll 


122862 


15.73 


996151 


.28 


126211 


16-01 


878789 


23 


128806 


15-69 


996184 


.28 


127172 


15.97 


872828 


22 


39 


124248 


i5.66 


9961 17 


-28 


128180 


15.94 


871870 


21 


40 


125187 


15.62 


996100 


-28 


129087 


15-91 


8-70913 


20 


4i 


9-126125 


15.59 


9-996088 


-29 


9.180041 


15-87 


10-869959 


'.t 


42 


1 27060 


15.56 


996066 


-29 


180994 


15-84 


869006 


43 


127998 


i5.52 


996049 


-29 


181944 


i5-8i 


868o56 


\l 


44 


128525 


15.49 


996082 


-29 


182893 


i5-77 


867107 


45 


129854 


15.45 


996015 


.29 


133889 


15-74 


866161 


i5 


46 


180781 


15.42 


995908 


•29 


184784 


15.71 
15.67 


865216 


14 


s 


181706 
182680 


i5-39 
15.35 


995980 


•29 


185726 


864274 
863338 


i3 


993968 


.29 


186667 


i5-64 


la 


49 


i8355i 


1532 


995946 


.29 


187605 
188542 


i5.6i 


862895 
S61458 


II 


5o 


134470 


i5'29 


995928 


.29 


i5.58 


10 


5i 


9.135387 


i5-25 


9-995911 


.29 


9-189476 


15.55 


io.86o524 


t 


52 


186808 


15-22 


995894 


-29 


140409 


i5.5i 


859591 
858660 


53 


187216 
188128 


15.19 


995876 


•29 


I 41 840 


15.48 


I 


54 


i5.i6 


995359 


.29 


142269 


15.45 


857781 
8568o4 


55 


189087 


l5-12 


995841 


.29 


148196 


15.42 


5 


56 


139044 
i4o85o 


i5-09 


995828 


.29 


144121 


15.39 
i5-35 


855879 


4 


U 


i5-o6 


995806 


-29 


145044 


854956 


3 


141754 
142655 


i5-o3 


995788 


-29 


145966 


i5-32 


854034 


a 


55 


15-00 


995771 


-29 


146885 


15-29 


853ii5 


I 


60 


143555 


14.96 


995753 


.29 


147803 


i5-26 


852197 



M. 


Gofiine 


D. 1 


Sine 




Ootan^. 


D. 


Tiui^. 








(82 


DBGB 


EBB.) 









86 


kS 


DEGREES.) A 


fABLE OF LOGAlilTHMlC 




M. 


Sine 


D. 


Cosine 


D. 


Tang. 


! ^^ 


Cot&Dg. 


r"^ 


o 


9-143555 


14-96 


9-995753 


•3o 


9 -147503 


15^26 


:o-853i97 


60 


1 


144453 


14-98 


990735 


-3o 


148718 


15-28 


831282 59 
85o368 58 


a 


i4534<) 


14-90 


993717 


-3o 


149682 


l5-20 


3 


146243 


14-87 


990699 


-3o 


i3o544 


if-17 


849436 


57 


4 


I47i36 


14-84 


995681 


• 3o 


i5i454 


i5-i4 


848546 


56 


5 


148026 


i4-8i 


995664 


-3o 


152363 


i5-ii 


847687 


55 


6 


14801 5 
149002 


14-78 


995646 


-3o 


153269 


i5.o8 


846781 


54 


I 


14-75 


995628 


-3o 


154174 


i5.o5 


845826 


53 


i5o686 


14-72 


995610 


-3o 


1 55077 


l5-02 


844928 


52 


9 


r5i569 


14-69 


993591 


-3o 


155978 


14-99 


844022 


5i 


10 


1 5245 1 


14-66 


995573 


•3o 


156877 


14-96 


848128 


5o 


i> 


9' I J33o 


14-63 


9 •995555 


.30 


9-157775 


14-93 


10-842225 


49 


12 


1 54208 


14-60 


995537 


.30 


108671 


U^oo 
14-87 


841329 


48 


i3 


i55o83 


14-57 


995519 


.80 


159565 


840435 


47 


U 


i55o57 
I 56830 


14-54 


995301 


.81 


160457 


14.84 


889548 
888653 


46 


i5 


i4-5i 


995482 


-81 


161847 


14-81 


45 


i6 


157700 
I 58569 


14-48 


995464 


•81 


162286 


14.79 


887764 


44 


\l 


14-45 


993446 


•31 


168128 


14.76 


886877 


43 


i5943d 


14-42 


993427 


.81 


164008 


14-73 


835992 


42 


«9 


i6o3oi 


14-39 


993409 


•3i 


164892 


14.70 


835 108 


41 


20 


161 164 


14-36 


993890 


■3i 


165774 


14.67 


884226 


40 


21 


9' 162025 


14-38 


9-995372 


• 31 


9- 166634 


14-64 


10-833346 


39 

38 


22 


162885 


i4'8o 


995358 


.81 


167532 


i4-6i 


882468 


S3 


168743 


14-27 


995334 


• 3i 


168409 


14-58 


881591 


J7 


24 


164600 


14-24 


993316 


• 31 


169284 


I4-35 


880716 


36 


25 


165454 


14-22 


993297 


-31 


170157 


14-53 


829843 


35 


26 


i663o7 


14-19 


993278 


• 31 


171029 


i4-5o 


828971 


34 


11 


i67i59 


14- 16 


995260 


'i' 


171809 
172767 


14-47 


828101 


83 


168008 


i4-i8 


995241 


.32 


14-44 


827288 


82 


29 


168856 


14-10 


995222 


•32 


178684 


14-42 


826866 


81 


3o 


169702 


14-07 


995208 


.82 


174499 


14-89 


825501 


3o 


3i 


9-170547 


i4-o5 


9-995184 


.82 


9-175362 


14-36 


10-824688 


^ 


33 


171889 


14-02 


9951 65 


.82 


176224 


14-33 


828776 


33 


172280 


18-99 


995146 


.82 


177084 


i4-3i 


822916 


27 


34 


178070 


13-96 


995127 


.82 


177942 


14-28 


822058 


26 


35 


178908 


i3-94 


995108 


.82 


178799 


14-25 


821201 


25 


36 


174744 


18-91 


995089 


•32 


179655 


14-28 


820845 


24 


U 


175578 


13-88 


993070 


.32 


i8o5o8 


14-20 


819492 
818640 


28 


176411 


13-86 


99305 I 


.82 


i8i36o 


14-17 


22 


39 


177242 


12-88 


Q93032 


82 


182211 


14.15 


817789 


21 


4o 


178072 


i3-8o 


995oi3 


.32 


188009 


14.12 


816941 


20 


41 


9-178900 


18-77 


9-994993 


.82 


9-188907 


14.09 


10-816098 


;? 


43 


179726 


18.74 


994974 


.82 


184752 


14-07 


815248 


43 


i8o55i 


18.72 ! 


994935 


•32 


185597 


14-04 


814408 


n 


44 1 


18187.' 


18.69 


994935 


•32 


186439 


14-02 


8i356i 


16 


45! 


182196 


13-66 


994916 


.33 


187280 


18-99 


812720 


i5 


46 , 


i83oi6 


i3-64 


994896 


•33 


188120 


18-96 


811880 


14 


8 


183884 


18-61 i 


994877 


■88 


188958 


18-98 


811042 


i3 


i3465i 


18-5^ , 


994857 


-83 


189794 


18-91 
18-89 


810206 


" ; 


49 


185466 


i3.5b ; 


994838 


.83 i 


190629 


800871 ; 


11 


te 


186280 


i3-:>3 


994818 


•33 


191462 


18-86 


8o6588 


10 


5i 


9- 10-092 


i3.5i 


9-994798 


•88 


9-192294 


18-84 


10-807706 


t 


5> 


187908 


13.48 1 


9947-^9 


.33 


198124 


:3-8i 


806876 


53 


188712 1 


18-46 


994759 


.88 


198953 


18-79 


806047 


I 


54 


i8o5i9 1 


13-48 


994789 


33 


194780 j 


13-76 ., 


8o5220 


55 


190823 1 


18.41 


994719 


33 


193606 


13-74 


804394 


5 


56 


191180 


i3-38 


994700 


88 


196480 


18-71 


808570 


4 


57 


191988 1 


13-86 


994680 


88 


197253 


18-69 


802747 


3 


X 


192784 ! 


18.88 ! 


994660 


38 


198074 


i3-66 


801926 


2 


55 


198534 ' 


i3-3o 


c-4640 


•33 


19S894 1 


18-64 


801106 


I 


66 


194382 i 


18-28 


:g,"20 


• 33 


199718 j 


i3-6i 


800287 . 





1 


Coftine j 


D. 1 


Sine 


, 


Cotan^. ' 


D. t 


Tanff. 








(81] 


DEGR] 


EB£.) 









i 





SINES AND TANGENTS. (9 DEGREE.') 




27 


I'm. 


Bine 


D. 


Ooeine 


D. 


Tang., 


D. 


Cotang. 







9-194333 


i3-28 


9.994620 


.33 


9.199713 


i3-6i 


10-800287 


60 


I 

a 


195129 
195920 


i3-26 
i3-23 


994600 
994580 


.33 
• 33 


200520 

201845 


13.59 
13-56 


799471 
798655 


u 


3 


196719 


13-21 


994560 


.34 


202159 


13-54 


797841 


57 


4 


197511 


i3-i8 


994540 


.34 


202971 


i3-52 


797029 


56 


5 


198302 


13-16 


994519 


.34 


208782 


18.49 


796218 


55 


6 


I 9909 I 


i3.i3 


994499 


•?4 


204592 


13-47 


795408 


54 


I 


199879 


13-11 


994479 


.34 


205400 


i3-45 


7q46oo 


53 


200666 


i3-o8 


994456 


.34 


206207 


18-42 


798798 


52 


9 


20i45i 


13-06 


994438 


-34 


207018 


13.40 


792987 


5i 


10 


202234 


i3-o4 


994418 


-34 


207817 


13-88 


792183 


5o 


II 


9«2o3oi7 


i3-oi 


9-994397 


-34 


9-208619 


13-85 


10-791881 


49 


12 


203797 
204577 
205354 


12-99 


994377 


.34 


209420 


18.33 


790580 


48 


i3 


12-96 


994357 


.34 


210220 


13. 3i 


780780 


47 


U 


12-94 


994336 


-84 


21IO18 


18.28 


788982 


46 


15 


2061 3 I 


12-02 


9943x6 


-34 


2ii8i5 


18^26 


788185 


45 


16 


206906 


12-89 


994295 


•34 


2I26I1 


i3^24 


787889 


44 


;? 


207679 


12.87 


994274 


•85 


2i34q5 


18^21 


786596 


43 


208452 


12-85 


994254 


-85 


214198 


i3'i9 


785802 


42 


I? 


209222 


12-82 


994233 


-85 


214989 


18-17 


78501 1 


41 


20 


209992 


12-80 


994212 


.35 


215780 


i8-i5 


784220 


40 


21 


9-210760 

211326 


12.78 


9.994191 


.35 


9-216568 


l8-I2 


10.788482 


^2 


22 


12-75 


9941 7 1 


-35 


217856 


i3.io 


782644 


38 


23 


212291 


12-73 


9941 5o 


-35 


218142 


18.08 


78x858 


37 


24 


2i3o55 


12.71 


994120 


-35 


218926 


i3.o5 


781074 


86 


25 


2i38i8 


12.68 


994108 


.35 


219710 


i3.o8 


780290 


85 


26 

11 


214579 
215338 


12-66 
12-64 


994087 
994066 


.35 
.35 


220492 
221272 


18.01 
12-99 


77q5o8 
778728 


34 
33 


216097 


12-61 


994045 


.35 


222052 


12-97 


777948 


32 


29 


216854 


12-59 


994024 


.35 


222880 


12-94 


777170 


3i 


3o 


217609 


12-57 


994oo3 


-35 


228606 


12.92 


776394 


3o 


3i 


9-218363 


12.55 


9.998981 


-35 


9-224882 


12-90 


10-7756x8 


ll 


32 


219116 


12.53 


998960 


.35 


225i56 


12-88 


774844 


33 


219868 


12'5o 


998939 


-35 


225929 


12-86 


774071 


27 


34 


220618 


12.48 


998918 


-35 


226700 


12-84 


773300 


26 


35 


22i367 

3221l5 


12.46 


998896 


-36 


227471 


12-81 


772529 


25 


36 


12.44 


99G875 


-36 


228289 


12.79 


771761 


24 


ll 


222861 


12-42 


998854 


-36 


229007 


12.77 


770998 


23 


223606 


12.39 


998882 


• 36 


229773 


12-75 


770227 


22 


39 


224349 


12.37 


998811 


.36 


280589 


12.78 


769461 
768698 


21 


40 


225092 


12.35 


998789 


-36 


281802 


12.71 


20 


41 


9-225833 


12-33 


9.998768 


-86 


9-282065 


12-69 


10-767935 


19 


42 


226573 


12-3l 


998746 


-86 


282826 


12-67 


767174 


18 


43 


227311 

22S048 


12-28 


998725 


-86 


283586 


12-65 


766414 


17 


44 


12-26 


998708 


-36 


284845 


12-62 


765655 


16 


45 


228784 


12-24 


998681 


-36 


285io3 


12-60 


764897 


i5 


46 


229518 


12-22 


998660 


-36 


285SJ9 


12-58 


764141 


14 


J 48 


230252 


12.20 


998688 


.36 


286614 


12.56 


768886 


i3 


230984 


12-18 


998616 


-36 


287868 


12-54 


762682 


12 


49 


231714 


12-16 


998594 


-37 


288120 


12-52 


761880 


11 


5o 


232/i/i/i 


12-14 


993572 


•37 


288872 


12 -5o 


761128 


TO 


5. 


9.233172 


12-12 


9.993550 


.37 


9-289622 


12.48 


10-760878 


? 


52 


233899 


12-09 


998528 


.37 


240871 


12-46 


759629 


53 


234626 


12-07 


993506 


-37 


241 1 18 


12-44 


758882 


I 


54 


23534Q 


12-05 


998484 


•37 


241 865 


12-42 


758i35 


55 


23607J 


12. o3 


998462 


•37 


242610 


12-40 


757890 


5 


56 


236795 


12-01 


998440 


•37 


248354 


12-38 


756646 1 4 


u 


2375i5 


11-99 


993418 


•37 


244097 
244889 


12-86 


755oo3 1 3 


238235 


11-97 

11.95 


998896 


.37 


12-34 


755161 2 


1 59 


238953 


993374 
993351 


•37 


245579 


12-32 


754421 I 


60 


239670 


11.93 


•37 


246819 


12-30 


753681 

1 


1 


Coeino 


D. 


Sine 


Cotang. 


D. 


Tang. 


M. 








(80 


DSOI 


IXSB.) 









28 



(10 DEGREES.) A TABLE OF LOGARITHMIC 



M. 




Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 


9-239670 
240386 


11.93 


9.993351 


.37 


9' 246819 


12-30 


10.753681 


I 


II'OI 


993329 


•37 


247067 


12.28 


762943 


5? 


2 


241 lOI 


11.89 


993307 


.37 


248530 


12.26 


762206 


3 


241814 


11.87 


993285 


.37 


12-24 


751470 
760736 


u 


4 


242626 


11-85 


993262 


•37 


249264 


12.22 


5 


243237 


11-83 


993240 


:U 


249998 
260780 


12.20 


760002 


55 


6 


243947 
244656 


11.81 


993217 


12.18 


749270 


54 


I 


11.79 


993196 


.38 


261461 


12-17 
12-l5 


748539 


53 


245363 


WJ,l 


993172 


• 38 


262191 


747809 


52 


9 


246069 
246775 


998149 


• 38 


262920 


1213 


747080 


5i 


10 


11.73 


993127 ' 


• 38 


268648 


12-11 


746862 


5o 


11 


9-247478 
248181 


11-71 


9.993104 


.38 


9.264874 


12-09 


10-745626 


S 


la 


11.69 


998081 


.88 


266100 


12-07 
12-05 


744900 


i3 


248883 


11.67 


998069 


.38 


266824 


744176 
743463 


% 


14 


249583 


11.65 


998086 


.38 


266647 


12-03 


i5 


250282 


11-63 


998013 


.38 


267269 


12.01 


742781 


45 


i6 


260980 


11-61 


992990 


.38 


267990 
268710 


12.00 


742010 


44 


\l 


25167-' 


11.59 


992967 


.38 


11.98 


741290 


43 


262373 


11.68 


992944 


.38 


269429 


11.96 


740671 


42 


19 


253067 


11.66 


992921 


.38 


260146 


11-94 


789864 


41 


20 


253761 


11-64 


992898 


.38 


260868 


11.92 


789137 


40 


21 


9.264453 


11-62 


9.992876 


.38 


9.261678 


11.90 


10-788422 


^ 


22 


256i44 


11-50 


992862 


.38 


262292 


11-89 


787708 


23 


255834 


11.48 


992829 


.39 


268006 


11-87 
11-85 


786996 


u 


24 


256623 


11-46 


992806 


.39 


268717 


786283 


25 


267211 


11.44 


992788 


.39 


264428 


11-83 


786672 


35 


26 


267898 


11.42 


992769 


.39 


265i38 


11-81 


784862 


34 


11 


258583 


11.41 


992786 


.39 


266847 


11.79 


784153 


33 


269268 


11.39 


992713 


.89 


266565 


11.78 


788445 


32 


29 


259961 


11.37 


992690 


.39 


267261 


11-76 


732739 
782033 


3i 


3o 


260633 


11-35 


992666 


.39 


267967 


11-74 


3o 


3i 


9-26i3i4 


11-33 


9-992648 


.39 


9.268671 


11-72 


10-781829 
780626 


It 


32 


261994 


11-31 


992619 


.39 


269876 


11-70 


33 


262673 
263351 


11-30 


992696 


19 


270077 


11-69 


729928 


ll 


34 


11-28 


992672 


.39 


270779 


11-67 
11-65 


729221 


35 


264027 
264703 
266377 


11.26 


992649 


.39 


271479 


720621 


25 


36 


11-24 


992626 


•39 


272178 


11-64 


727822 


24 


ll 


11-22 


992601 


.39 


272876 


11-62 


727124 


23 


266061 


11-20 


992478 


.40 


278678 


11-60 


726427 


22 


39 


266723 
267396 


11-19 


992464 


.40 


274269 


11-68 


726731 


21 


40 


11-17 


992480 


.40 


274964 


11-67 


726086 


20 


41 


9-268065 


ii-i5 


9.992406 


.40 


9.276668 


11.56 


10-724842 


\t 


42 


268734 


I1-I3 


992882 


.40 


276861 


11-63 


728649 


43 


269402 


11-11 


992869 


.40 


277043 


ii-5i 


722967 
722266 


\l 


44 


270060 
270736 


II-IO 


992835 


.40 


277784 
278424 


11-60 


45 


11.08 


99281 1 1 


.40 


11.48 


721676 
720887 


i5 


46 


271400 


11.06 


992287 1 


.40 


279118 


11-47 


14 


il 


272064 


11.06 


992268 


.40 


279801 
280488 


11.46 


720199 


i3 


272726 


II.03 


992289 


.40 


11.43 


719612 


12 


49 


273388 


11.01 


992214 


.40 


281174 


11.41 


718826 


II 


5o 


274049 


10.99 


992190 


.40 


281868 


11.40 


718142 


10 


5i 


9.274708 


10.98 


9.992166 


.40 


9.282542 


11.38 


10-717468 


I 


52 


275367 


10.96 


992142 


.40 


288226 


11.36 


716775 


53 


276024 


10.94 


992117 


•41 


288907 


11.35 


716093 


7 


54 


276681 


10.92 


992098 


-41 


284688 


11-33 


716412 1 6 1 


55 


277337 


10.91 


992069 


•41 


286268 


ii-3i 


71473a 


5 


56 


277991 
278644 


10.89 


992044 


•41 


286947 


ii-3o 


714063 


4 


u 


10.87 


992020 


•41 


286624 


11-28 


718876 


3 


279297 
279948 
280399 


10.86 


991996 


•41 


287801 


11-26 


712690 
712023 


2 


59 


10.84 


991971 


•41 


287977 


11-26 


I 


60 


10.82 


991947 


•41 


288662 


11-23 


711348 





- 


Ck>Bino 


D. 


Sine 


Cotang. 


D. 


Tang. 


M. 



(79 DEGREES.) 



SINKS AND TANGENTS. (11 DEGREES.) 



29 



M. 


Sine 


D. 


Coeine 


D. 


Tang. 


D. 


Cotan^. 







9 '280599 
281240 


10-82 


9 991947 


•41 


9-288652 


11-23 


10.711848 


60 


I 


10 


81 


991922 


•41 


289826 


11-22 


710674 


u 


) 


281897 


10 


79 


991897 
991878 


•41 


289999 


11-20 


710001 


3 


282544 


10 


11 


•41 


290671 


11-18 


709829 


u 


4 


288190 
283836 


10 


991848 


•41 


291842 


11-17 

ii-i5 


7o8658 


5 


10 


•74 


991828 


•41 


292018 


707987 


55 


6 


284480 


10 


72 


991799 


•41 


292682 


11-14 


707818 


54 


I 


285124 


10 


71 


991774 


•42 


298350 


11-12 


7o665o 


53 


385766 


10 


69 


991749 


•42 


294017 


ii-ii 


7o5q83 


52 


9 


286408 


10 


tl 


991724 


•42 


294684 


11-09 


7053 1 6 


5i 


10 


287048 


10 


991699 


•42 


295849 


11.07 


704651 


5o 


11 


9.287687 
288826 


10 


64 


9-991674 


•42 


9' 296018 


11-06 


10-708987 


49 

48 


12 


10 


63 


991649 


•42 


296677 


11-04 


703328 


i3 


288964 


10 


61 


991624 


•42 


297889 
298001 


II-03 


702661 


47 


14 


289600 


10 


5o 


991599 


•42 


11-01 


7°^?99 
701888 


46 


i5 


290286 


10 


58 


991574 


•42 


298662 


11-00 


45 


i6 


290870 


10 


56 


991549 


•42 


299822 


10.98 


700678 


44 


\l 


291504 


10 


54 


991524 


♦42 


299980 


10-96 


700020 
699862 
698705 


43 


292187 
292768 
298J99 


10 


53 


991498 


•42 


800688 


10-95 


42 


19 


10 


5i 


991473 


•42 


801295 


10-98 


41 


20 


10 


5o 


991448 


•42 


301961 


10-92 


698049 


40 


21 


9 '294029 


10 


48 


9.991422 


•42 


9-802607 


10-90 


10-697898 
696789 


^ 


22 


294658 


10 


46 


991897 


.42 


808261 


10-89 


23 


295286 


10 


45 


991372 


.43 


308914 
304367 
3o52i8 


10-87 
10-86 


696086 


U 


24 


295918 


10 


43 


991346 


.43 


695433 


25 


296539 


10 


42 


991821 


.43 


10-84 


694782 


35 


26 


297164 


10 


40 


991295 


•43 


3o5869 


10-83 


694181 


34 


11 


297788 


10 


39 


991270 


'43 


3o65i9 


10-81 


693481 


33 


298412 


10 


ll 


991244 


.43 


807168 


10-80 


692882 


32 


29 


299084 


10 


991218 


•43 


807815 


10-78 


692185 


3i 


3o 


299655 


10 


34 


991193 


.43 


808468 


10-77 


691537 


3o 


3i 


9.800276 


10 


32 


9'99ii67 


.43 


9*809109 


10-75 


10-690891 


It 


32 


800895 


10 


81 


991141 


.43 


809754 


10-74 


690246 


33 


3oi5i4 


10 


29 


991115 


.43 


810898 


10-78 


689602 
688958 

6883 1 5 


ll 


34 


802182 


10 


28 


991090 


.43 


811042 


10-71 


36 


802748 


10 


26 


991064 


.43 


3ii685 


10-70 
10-68 


25 


36 


808864 


10 


25 


991088 


.43 


812827 


687673 


24 


ll 


803979 
804598 


10 


23 


991012 


.43 


312967 
818608 


10-67 
10-65 


687088 
686892 
685753 


23 


10 


22 


990986 


•43 


22 


39 


3o5207 


10 


20 


990960 


•43 


314247 
814885 


10-64 


21 


40 


3o58i9 


10 


19 


990934 


•44 


10.62 


6851 i5 


20 


41 


9 '306430 


10 


17 


9 '990008 


•44 


9-3i5528 


10.61 


10-684477 


19 


42 


807041 


10 


16 


990882 


•44 


816159 


10-60 


688841 


18 


43 


807650 
808259 


10 


14 


990855 


•44 


816793 
817430 
818064 


10-58 


688205 


\l 


44 


10 


i3 


990829 
990808 


•44 


10-57 
10-55 


682570 


45 


808867 


10 


11 


•44 


681986 


i5 


46 


809474 
3 10000 


10 


10 


990777 


•44 


818697 


10-54 


68i3o3 


14 


ii 


10 


08 


990750 


•44 


819829 


10'53 


680671 


i3 


8io685 


10 


ll 


990724 


•44 


819961 


iO'5i 


680089 


12 


49 


811289 
811898 


10 


990697 


•44 


820592 


iO'5o 


679408 


11 


5c 


10 


04 


990671 


•44 


821222 


10-48 


678778 


10 


5i 


9-812495 


10 


o3 


9-990644 


•44 


9-82i85i 


i0'47 
10 '45 


10-678149 


t 


52 


818097 
818698 


10 


01 


990618 


•44 


822479 


677521 


53 


10 


00 


99o5qi 
99o5o5 


•44 


828106 


10.44 


676894 


I 


54 


314297 


9 


98 


•44 


328788 


10-43 


676267 


55 


314897 
815495 


9 


97 


990538 


•44 


324858 


10-41 


675642 ' 5 


56 


9 


96 


9905 11 


•45 


824983 


10-40 


675017 4 


5j 


816092 
816689 


9 


94 


990485 


•45 


825607 


10-89 


674893 


3 


58 


9 


93 


990458 


.45 


826281 


10-87 
10-36 


678769 


3 


59 


817284 


9 


91 


990481 


•45 


326853 


678147 
672525 


I 


6o 


817879 


9.90 


990434 


•45 


827475 


10-35 







CJosine 


D. 


Sine 




Cotan^. 


D. 


Tang. ^ 


M. 



(78 DBGRBEB.) 



■60 



(12 DEGREES.^ A TABLE OF LOGARITHMIC 



H. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotanfr. 




o 


9.317870 
318473 
319066 


9.§8 


9-990404 


•45 


9-327474 
328096 


10-35 


10-672526 


60 


I 


990878 
99085 I 


•46 


10.33 


671905 


n 


2 


9.87 


•46 


828716 


10-32 


671285 


3 


319658 


9-86 


990824 


-45 


829334 


io-3o 


670666 


U 


4 


320249 


9.84 


990297 


•45 


829963 
330670 
331187 
33i8o3 


10-29 


670047 


5 


320840 


0-83 


990270 


•45 


10-28 


669480 


55 


6 


321480 


9-82 


990248 


•45 


10-26 


668813 


54 


I 


322019 


9-8o 


990215 


•45 


10-26 


668107 


53 


322607 


9-79 


990188 


•45 


882418 


10-24 


667682 


5a 


9 

10 


328194 
328780 


9-77 
0-76 


990161 
990184 


•46 
•45 


383o33 
333646 


10-23 

10.21 


666967 
666354 


5i 
So 


It 


9 '824866 


9.75 


9-990107 


.46 


9-384269 


10-20 


10-665741 


^ 


12 


324950 
325534 


9.73 


990079 


-46 


884871 


10-19 


666129 
664618 


i3 


9.72 


990062 


•46 


336482 


10-17 


% 


14 


3261 17 


9.70 


990025 


.46 


336098 


10-16 


668907 
668298 
662689 


i5 


826700 


9.69 


989997 


.46 


886702 


io-i5 


45 


i6 


327281 


9.68 


989970 


•46 


337811 


io-i3 


44 


\l 


827862 
828442 


9.66 


989942 


.46 


337919 

33852-1 
339133 


10-12 


662081 


43 


9-65 


989915 


.46 


10-11 


661473 


42 


19 


829021 


9.64 


989887 


.46 


10-10 


660867 


41 


20 


829699 


9-62 


989860 


-46 


339789 


10-08 


660261 


40 


21 


9'88oi76 
330758 
331829 
3^1908 


9-61 


9-989882 


•46 


9-840844 


10-07 

10-00 


10-669666 


It 


22 


9-60 


989804 


.46 


340948 


669062 
668448 


23 


9-58 


989777 


.46 


341562 


10-04 


u 


24 


It 


989749 


•47 


842166 


10 -08 


667845 


25 


332478 
338o5i 


989721 


•47 


342767 


10-02 


667243 


35 


26 


9.54 


989698 


•47 


343358 


10-00 


656642 


34 


11 


338624 


9.53 


989665 


•47 


843968 


9.90 

9.98 


666042 


33 


334195 


9-52 


989687 


•47 


344568 


666442 


3a 


29 


334766 


9.50 


989609 


•47 


346167 
345766 


9-97 
9.96 


664843 


3i 


3o 


335837 


9.49 


989682 


•47 


664246 


3o 


3i 


9 •335906 


9.48 


9.989663 


•47 


9 •346353 


9.94 


10-653647 


S 


32 


886475 


9.46 


989626 


•47 


346949 
347545 


9-98 


668061 


33 


337048 


9.45 


989497 


•47 


9-92 


662455 


11 


34 


337610 
888176 


9.44 


989469 


•47 


348141 


9-91 


661869 


35 


9.43 


989441 


•47 


348735 


9-92 


651265 


25 


36 


338742 
389806 


9-41 


989418 


•47 


849829 


9-88 


660671 


24 


U 


9.40 


989884 


•47 


349922 


riz 


660078 
649486 


23 


339871 


9.89 


989866 


•47 


85o5i4 


22 


39 


340434 


VM 


089828 


•47 


861106 


9-85 


648894 


21 


40 


340996 


989800 


•47 


361697 


9-83 


648808 


20 


41 


9-34i558 


9.35 


9-989271 


•47 


9-862287 
352876 
853465 


9-82 


10.647718 


\t 


42 


342119 


9.34 


989243 


•47 


9-81 


647124 


43 


342679 


9-82 


989214 


•47 


9-80 


646535 


»7 


44 


348289 


9.81 


989186 1 


•47 


864063 


9-79 


646947 


6 


45 


343797 
344355 


9'8o 


969167 ' 
989128 


:il 


864640 


9-77 
9.76 


645360 1 5 1 


46 


9.29 


356227 
3558i3 


644773 
644107 


14 


S 


344912 


9.27 


989100 


.48 


9-75 


i3 


345469 


9-26 


989071 


.48 


366898 
866982 
357666 


9-74 


648602 


i* 


49 


846024 


9-25 


989042 


-48 


9-78 


648018 


II 


5o 


346679 


9-24 


989014 


.48 


9.71 


642434 


10 


5i 


9.347134 


^.22 


9.988985 


.48 


9.368149 


9-70 


10-641861 


t 


52 


347687 


9-21 


988966 


.48 


868781 


9-09 


641269 


53 


348240 


9'20 


988927 
988898 


.48 


869313 


9.55 


640687 


I 


54 


348792 
349843 


9.19 


.48 


369898 


|:^ 


640107 
689526 
688047 
638366 


55 


9.17 

9.16 


988869 


.48 


860474 


5 


56 


349898 


988840 


.48 


861068 


9-65 


4 


U 


350443 


9*15 


988811 


•49 


861682 


9-63 


3 


350992 
35 1 540 


9-14 


988782 


•49 


862210 


9-62 


637790 


a 


5q 


9.18 


988768 


•49 


862787 
363364 


9-61 


687213 
636636 


I 


66 


352088 


9-11 


988724 


.49 


9-60 







Cosine 


D. 


Sine 




Cotang. 


D. 1 


T«ng. _ 


Jkl 



(77 DXOR8X8.) 



StPTES AND TANGENTS. (13 DEGREES.) 



SI 






Sino 


D. 


Cosine 


D. 


Ttog, 


D. 


Cotang. 


60 


'» 352088 


9-11 


Q. 988724 


-49 


9-363364 


9-60 


10-636636 


I 


352635 


9 


10 


988695 


-49 


363940 
3645 I 5 


9-59 


636o6o 


U 


2 


353i8i 


9 


S 


988666 


.49 


9-58 


635485 


3 


353726 


9 


988636 


.49 


365o90 


^12 

9-55 


634910 
634336 


U 


4 


354271 


9 


:S 


988607 


.49 


365664 


5 


354815 


9 


988578 


.49 


366237 


9.54 


633763 


55 


e 


355358 


9 


04 


988548 


.49 


366810 


9.53 


633190 


54 


I 


355901 


9 


o3 


988519 


•49 


367382 


9.52 


632618 


53 


356443 


9 


02 


988489 


.49 


367953 


9.51 


632047 
631470 


52 


9 


356984 


t 


01 


988460 


.49 


368524 


9.50 


5i 


10 


357024 


99 


988430 


-49 


369094 


9.49 


630906 


5o 


II 


o> 358064 


8 


98 


9.988401 


•49 


9.369663 


9.48 


io.63o337 
629768 


it 


12 


3586o3 


8 


97 


988371 


.49 


370232 


9-46 


i3 


359141 


8 


96 


988342 


.49 


370799 


9.45 


629201 


47 


14 


359678 


8 


95 


988312 


.50 


371367 


9.44 


628633 


46 


i5 


36o2i5 


8 


93 


988282 


-5o 


371933 


9-43 


628067 


45 


i6 


360752 


8 


92 


988252 


-5o 


372499 


9.42 


627501 


44 


\l 


361287 


8 


91 


988223 


-5o 


373c64 


9-41 


626936 
626371 


43 


361822 


8 


90 


988193 


-5o 


373629 


9.40 


42 


«9 


362356 


8 


89 


988163 


-50 


374193 
374756 


9.30 


625807 


41 


20 


362889 


8 


88 


988133 


-5o 


9.3^ 


625244 


40 


21 


9-363422 


8 


87 
85 


9-988103 


-5o 


9.375319 


9.37 


10-624681 


39 


22 


363954 


8 


988073 


-5o 


375881 


9.35 


6241 19 


38 


23 


364485 


8 


84 


988043 


-5o 


376442 


9.34 


623558 


37 


24 


365oi6 


8 


83 


988013 


.5o 


377003 


9.33 


622997 
622437 


36 


25 


365546 


8 


82 


987983 


-5o 


377563 


9.32 


35 


26 


366075 


8 


81 


987953 


-5o 


378122 


9.31 


621878 


34 


11 


366604 


8 


80 


987522 


-5o 


378681 


9-30 


621319 


33 


367131 


8 


79 


987892 


-5o 


379239 


9.20 


620761 


32 


?9 


367659 
368i85 


8 


77 


987862 


-5o 


379797 


9.28 


620203 


3i 


3o 


8 


76 


987832 


-5i 


38o354 


9-27 


619646 


3o 


3i 


9'3687ii 


8 


75 


9-987801 


.51 


9.380910 


9. 20 


10-619090 


It 


32 


369236 


8 


74 


987771 


-5i 


38 1466 


9-25 


618534 


33 


369761 


8 


73 


987740 


-5i 


382020 


9-24 


617980 


ll 


34 


370285 


8 


72 


987710 


.51 


382575 


9-23 


617425 
616871 


35 


370808 


8 


71 


987679 


-5i 


383129 


9-22 


25 


36 


371330 


8 


70 


987640 
987618 


-5i 


383682 


9-21 


6i63i8 


24 


ll 


371852 


8 


69 


-51 


384234 


9.20 


615766 


23 


372373 


8 


tl 


987588 


-5i 


384786 
385337 
385888 


9-lQ 


6i52i4 


22 


39 


372894 


8 


987557 


-5i 


9.18 


614663 


21 


40 


373414 


8. 


65 


987526 


-5i 


9.17 


614112 


20 


41 


9-373933 


8. 


64 


9-987496 


• 51 


9.386438 


9-15 


io-6i3562 


\t 


42 


374452 


8 


63 


987465 


.5i 


386987 
387536 
388084 


9-14 


6i3oi3 


43 


374970 
375487 
376003 


8 


62 


987434 


.5i 


9-13 


612464 


\l 


44 


8 


61 


987403 


.52 


9-12 


611916 
611369 


45 


8 


60 


987372 


.52 


388631 


9-11 


i5 


46 


376519 
377035 


8 


U 


987341 


-52 


389178 


9-10 


610822 


14 


S 


8 


987310 


-52 


389724 


9.0Q 


610276 
609730 


i3 


377549 
378063 


8 


57 


987270 


-52 


390270 


9.08 


12 


49 


8 


56 


987248 


-52 


390815 


9.07 


609185 
608640 


11 


5o 


378577 


8 


54 


987217 


-52 


391360 


9.06 


10 


Si 


9-379089 


8 


53 


9.987186 


-52 


9.391903 


9-05 


10-608097 
607553 


? 


52 


379601 


8 


52 


987155 


-52 


392447 


9.04 


53 


38oii3 


8 


5i 


987124 


.52 


392989 


9.03 


60701 1 


I 


54 


380624 


8 


5o 


987092 


.52 


393531 


9.02 


606469 


55 


i8ii34 


8 


49 


987061 


•52 


394073 


9.01 


605927 
6o5386 


5 


56 


381643 


8 


48 


987030 


.52 


394614 


9.00 

h 


4 


t2 


382152 


8 


% 


986998 


-52 


396154 


604846 


3 


382661 


8 


986967 


•52 


395694 
396233 


6o43o6 


2 


59 


383 168 


8 


45 


986936 


•52 


tu 


603767 


1 


60 


383675 


8-44 


986904 


.52 


396771 


603229 





1 Cosine 


D. 


Sine 


1 CotADg. 


D. 


Tansr. 


JL_ 



^ 



(76 DEOKEES.) 



32 


(14 


DEGREES.; A lAJiLE OF LOaARITTrVfTU 




M. 


Sire 


D. 


Coaiii« 


D 


Tang. 


D. 


Cotang. ! 


9-333675 


8-44 


9 • 9S6004 

9.S-6.373 


52 


9-396771 


8.96 


10-603339 60 




384182 


8-43 


53 


397309 


8 


.96 


602691 5o 




3846.87 


8-42 


9S6.341 


53 


397846 


8 


.90 


602104 58 


1 4 


380192 
385697 


8-41 
8-40 


986809 

9.S6773 


.53 
■ 53 


398383 
398919 


8 
8 


•94 
.93 


601617 57 1 
601081 56 ■ 


1 J 


386201 


8-39 


986746 


53 


399450 


8 


.92 


600045 ; 55 , 


f 6 


386704 


8-33 


956714 


53 


399990 


8 


.91 


600010 54 


I 


387207 
387709 


8.37 
8-36 


9S6683 
986601 


53 
■53 


400024 
401 o58 


8 
8 




599476 i 53 ^ 
598942 5a 1 


- 9. 


388210 


8.35 


986619 


53 


401591 


B 


■ 80 


598409 ! 5i 
597876 ! 5o 


lO 


38871 I 


8.34 


986087 1 


53 


402124 


8 


.87 


II 


9.38921 1 


8-33 


9-986555 


53 


9-402656 


8 


.86 


10.597344 49 

596813 3 


13 


38971 1 


8-32 


986523 


■ 53 


403187 


8 


.85 


i3 


390210 


8.3i 


936491 


53 


403718 


8 


.84 


096282 47 


14 


390708 


8.3o 


936409 


.53 


404349 


8 


.83 


590^51 46 


i5 


391206 


8.28 


986427 


53 


404778 


8 


.82 


595222 45 


16 


391703 


8.27 


936395 


53 


4o53o-8 


8 


•81 


594692 44 


17 


392199 


8.26 


986363 


54 


405836 


8 


■80 


594164 43 


i8 


392690 


8.25 


986331 


54 


406364 


8 


■78 


593636 42 


19 


393191 
393685 


8.24 


986299 1 


54 


406892 


8 


593108 41 , 


30 


8.23 


986266 


54 


407419 


8 


■77 


592581 40 1 


21 


9-3941-9 


8.22 


9.986234 


54 


9.407945 


8 


■76 


10- 592000 3o 
091520 38 
591003 3] 


33 


394673 


8.21 


986202 


54 


408471 


8 


•75 


23 


390166 


8.20 


986169 


04 


408997 


8 


74 


24 


390608 


8.19 


986137 


54 


409021 


8 


74 


25 


396100 


8.18 


986104 


04 


410045 


8 


73 


589950 35 


36 


396641 


8.17 


936072 


54 


410569 


8 


72 


58q43i 34 


u 


397132 


8.17 


986039 


54 


41 1092 


8 


71 


588908 33 


397621 


8.16 


986007 


54 


4ii6i5 


8 


70 


588385 I 32 1 


29 


3981 I I 


8-15 


985974 


54 


412137 


8 


^ 


587863 1 3i 


3o 


398600 


8.14 


98594a 


54 


412608 


8 


587342 i 3o 


3i 

32 


9.399088 


8.i3 

8-12 


^■&? 


55 
55 


9.413179 
413699 


8 
8 


H 


10.586821 29 
586301 28 


33 


400062 


8. II 


985843 


55 


414210 


8 


65 


585781 27 


34 


400049 


8-10 


98581 I 1 


55 


414738 


8 


64 


585262 26 


35 


4oio3o 


8.09 
8.08 


985778 ' 


55 


415257 


8 


H 


584743 20 


36 


401020 


985745 


55 


415775 


8 


63 


584225 24 


U 


4o2oo5 


8.07 


980712 1 


55 


416293 


8 


62 


583707 23 


402489 


8.06 


980679 1 


55 


416810 


8 


61 


583i90 22 


39 


402972 


8.00 


985646 


55 


417326 


8 


60 


582674 21 


4o 


4o34o5 


8.04 


980613 


55 


417842 


8 


59 


582158 ao , 


41 


9-403938 


8-o3 


9 ■980080 ', 


55 


9-418358 


8- 


58 


10.58x642 


19 
18 


42 


404420 


8-02 


935547 i 


55 


418873 


8 


U 


581127 
58o6i3 


43 


404901 
400382 


8-01 


985514 


55 


419387 


8 


17 ' 


44 


8.00 


985480 


55 


419901 


8 


55 


580099 16 


45 


400862 


7-p 


985447 


55 


42041 5 


8 


55 


5795S0 i5 


46 


406341 


985414 


56 


420927 


8 


54 


579073 1 4 
578,^60 i i3 i 


S 


406820 


7-97 


985380 


56 


431440 


8 


53 


407299 


7.96 


985347 j 


56 


431953 


8 


5a 


578048 1 n I 


49 


407777 


7-95 


985314 ' 


56 


432463 


8 


5i 


577537 ' 11 i 


5o 


408204 


7-94 


980280 


56 


422974 


8 


5o 


577026 1 10 1 


5i 


9.4^731 


7-94 


9-985247 


06 


9.423484 


8 


S 


I0'5765i6 5 1 


52 


409207 


7.93 


9302 1 3 


06 


423993 


S 


576007 


8 


53 


409682 


7.92 


935180 


56 


434003 


8 


48 


075407 
574989 


I 


54 


410107 


7.91 


985146 


56 


435oii 


8 


% 


5t 


4io632 




985ii3 


56 


435519 


8 


574481 


5 1 


56 


41 1 106 


7.89 


980079 


56 


436037 


8 


45 


573973 




57 


411579 


7.88 


935040 


56 


426534 


8 


44 


573466 




5d 


41 3002 


7.87 


98501 I 


56 


437041 


8 


43 


572959 
572453 
571948 




S 


412024 
412996 


7.8^ 
7-85 


984978 
984944 


56 
56 


437547 
428003 


8 
8. 


43 
A2 


1 




Coeme 


D. 


:?ine 




Ccl&ng. 


D. 


Tar«. ilL i 








t'To Dl 


EGR 


EES.) 











I 



SINES AND TANGENTS. (15 DEGREES.; 



as 



M. 


Sina 


D. 


Ootiine 


D. 


Tanff. 


D. 


Cotang. 







9 412996 


7-85 


9-984944 


'i^ 


9.428052 


8.42 


10-571948 


60 


I 


4i34J'' 


7-84 


984910 


.57 


428557 


8.41 


571443 


5? 


3 


41393d 


7-83 


984876 


•57 


429062 


8.40 


570938 


3 


414408 


7-83 


984842 


•57 


429566 


8.39 
8.38 


570434 


57 


4 


414878 


7.82 


984808 


17 


430070 


569930 


56 


5 


415347 
4i58i5 


7.81 


984774 


.57 


430573 


8.38 


56q427 


55 


6 


7.80 


984740 


•57 


431075 


B.37 


568925 


54 


I 


416283 


7.70 


984706 


17 


43 1577 


3.36 


568423 


53 


416751 


7.78 


984672 


17 


432079 


8.35 


567921 


52 


9 


417217 


l]l 


984637 
984603 


.57 


432580 


8.34 


567420 


5i 


10 


417684 


•57 


433080 


8.33 


566920 


5o 


II 


9'4i8i5o 


7.75 


9.984569 


•57 


9.433580 


8.32 


10-566420 


49 


la 


4i86i5 


7-74 


984535 


17 


434080 


8.32 


565920 


48 


i3 


419079 


7-73 


984500 


17 


434570 


8.3i 


565421 


% 


14 


419544 


7.73 


984466 


•57 
.58 


435078 


8.3o 


564922 


i5 


420007 


7-72 


984432 


435576 


8.29 


564424 


45 


i6 


420470 


7-71 


984397 


.58 


436073 


8.28 


563927 


44 


\l 


420933 
421395 


]:Z 


984363 


.58 


436570 


8.28 


563430 


43 


984328 


.58 


437067 


8-27 
8-26 


562933 


42 


19 


421857 
4223i8 


7-68 


984294 


• 58 


437563 


562437 


41 


20 


7-67 


984259 


.58 


438059 


8-25 


561941 


40 


21 


9-422778 


]t 


9.984224 


.58 


9-438554 


8.24 


10.561446 


39 


22 


423238 


984190 


.58 


439048 


8-23 


560952 


38 


23 


423697 


7-65 


984155 


• 58 


439543 


8.23 


560457 


37 


24 


424106 


7-64 


984120 


.58 


44oo36 


8-22 


559964 


36 


25 


424615 


7-63 


984085 


.58 


44o529 


8-21 


559471 


35 


26 


425073 


7-62 


984050 


.58 


441022 


8.20 


558978 
558486 


34 


U 


425530 


7.61 


984015 


.58 


44i5i4 


8.19 


33 


425987 


7 -60 


983981 


• 58 


442006 


8.19 


557994 


32 


\9 


426443 


7-6o 


983946 


.58 


442497 


8.18 


557303 


3i 


3o 


426899 


7-59 


98391 I 


.58 


442988 


8.17 


557012 


3o 


3i 


9-427354 


7-58 


9.983875 


.58 


9-443479 


8.16 


10-556521 


!§ 


32 


427809 
428263 


]t 


983840 


.59 


443968 


8.16 


556o32 


33 


983805 


.59 


44/1458 


8.i5 


555542 


27 


34 


428717 


7-55 


983770 


.59 


444947 


8-14 


555o53 


26 


35 


429170 


7-54 


983735 


.59 


445435 


8-13 


554565 


25 


36 


429623 


7-53 


983700 


.59 


445923 


8-12 


554077 


24 


\L 


430075 


7-52 


983664 


.59 


44641 1 


8.12 


553589 


23 


43o527 
430978 


7-52 


983629 


.59 


446898 


8.11 


553102 


22 


39 


7-5i 


983594 


.59 


447384 


8-10 


552616 


21 


40 


431429 


7.50 


983558 


.59 


447870 


8.09 


552i3o 


30 


41 


9.431879 


7-49 


9.983523 


.59 


9.448356 


8.09 
8.08 


io.55i644 


\% 


42 


43232Q 


7-49 


983487 


.59 


448841 


55ir59 


43 


432778 


7.48 


983452 


.59 


449326 


8.07 


550674 


^1 


44 


433226 


7-47 


983416 


.59 


449810 


8.06 


550190 


16 


45 


433675 


7-46 


983381 


.59 


450294 


8.06 


549706 


i5 


46 


434122 


7.45 


983345 


.59 


450777 


8.o5 


540223 


14 


% 


434569 


7-44 


983309 
983273 


.59 


451260 


8.04 


548740 


i3 


435oi6 


1-U 


.60 


451743 


8.o3 


548257 
547773 


13 


i^ 


435462 


7-43 


983238 


.60 


452225 


8.02 


II 


5o 


435908 


7.42 


983202 


.60 


452706 


8-02 


547294 


10 


5i 


0-436353 


7-41 


9.983166 


.60 


9.453187 

453668 


8-01 


10.546813 


t 


5s 


436798 


7-40 


983 i3o 


.60 


8.00 


546332 


53 


437242 


7-40 


983094 


.60 


454 14S 


7.99 


545852 


I 


54 


437686 
438129 


7.39 


983o58 


.60 


454628 


7.99 


545372 


55 


7-38 


983022 


.60 


455107 


7.98 


544893 


5 


56 


438572 


7.37 


982986 


.60 


455586 


7-97 


544414 


4 


% 


439014 


7-36 


982950 


.60 


456064 


7-96 


543936 


3 


439456 


7-36 


982914 


.60 


456542 


7.96 


543458 


3 


^ 


439807 
440338 


7-35 


982878 


.60 


457019 


7-95 


542981 
542504 


1 


6o 


7-34 


982842 


.60 


457496 


7-94 





1 


Cosine 


D. 


Sine 




Cotaug. 


i D. 


Tang. 


M. 



(74 DSGRXBS.) 



S4 


C16 


DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 




8ino 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 


9-440338 


7-34 


9-982842 


-60 


j 9-457496 


7-94 


10 • 542504 


I 


440778 


7-33 


982805 


.60 


457973 


7-93 


542027 


5I 


a 


441218 


7-32 


982769 
982733 


-61 


458449 


7-93 


54i55i 


3 


441658 


7-3i 


-61 


458923 


7-92 


541075 


U 


4 


442096 
442535 


7-3i 


982696 


.61 


! 459400 


7-91 


C4000U 


5 


7-3o 


982660 


•61 


' 459875 


7.90 


540125 


5t> 


6 


442973 


7-29 


982624 


-61 


j 460349 


7-00 


539651 


54 


i 


443410 


7-28 


982587 


•61 


! 460823 


7-09 


539177 


53 


443847 


7-27 


982551 


•61 


: 461297 


7^88 


538703 


5a 


9 


444284 


]:ll 


982514 


-61 


! 461770 


7-88 


538230 


5i 


10 


444720 


982477 


.61 


462242 


7.87 


537758 


5o 


II 


9-445i55 


7-25 


9-982441 


.61 


9-462714 


7.86 


10-537286 


49 


12 


445590 


7-24 


982404 


.61 


! 463 186 


7-85 


536814 


48 


i3 


446025 


7-23 


982367 


-61 


463658 


7-85 


536342 


47 


14 


446459 


7-23 


982331 


.61 


1 464129 


7-84 


535871 


46 


i5 


446893 


7-22 


982254 


.61 


464599 


7-83 


535401 


45 


i6 


447326 


7-21 


982237 


•61 


i 465069 


7-83 


534931 


44 


17 


447759 
448191 


7-20 


982220 


•62 


1 465530 


7-82 


534461 


43 


i8 


7-20 


982183 


.62 


466008 


7-81 


533992 


42 


19 


448623 


7-19 


982146 


•62 


466476 


7-80 


533524 


41 


20 


449054 


7-18 


982109 


-62 


466945 


7-80 


533o55 


40 


21 


9-449485 


IM 


9-982072 


-62 


9-467413 


7-79 


10.532587 


39 


22 


449915 
45o345 


982035 


•62 


467880 


7-78 


532120 


38 


23 


7-16 


981998 


-62 


468347 


7-78 


53 1 653 


37 


24 


450775 


7-15 


981961 


-62 


468814 


7-77 


531186 


36 


25 


45 1 204 


7-14 


981924 


•62 


469280 


7-76 


530720 


35 


26 


45i632 


7-i3 


981886 


-62 


469746 


7-75 


530254 


34 


27 


452060 


7-i3 


981849 


-62 


470211 


7-75 


529789 
529324 


33 


28 


452488 


7-12 


981812 


-62 


470676 


7-74 


32 


29 


452915 


7-11 


981774 


-62 


471 141 


7-73 


528859 
528395 


3i 


3o 


453342 


7-10 


981737 


-62 


471605 


7-73 


3o 


3i 


9-453768 


7-10 


9-981699 


-63 


9-472068 


7-72 


10.527932 


ll 


32 


454194 


?:S 


981662 


-63 


472532 


7-71 


527468 


33 


454619 


981625 


•63 


472995 


7-71 


527005 


27 


34 


455o44 


7-07 


981387 


•63 


473437 


7-70 


526543 


26 


35 


455469 
455893 


7-07 


981549 


•63 


473919 


7.69 


526081 


25 


36 


7.06 


981512 


-63 


474381 


7-69 


5256iq 


24 


ll 


4563 1 6 


7-05 


981474 


•63 


474842 


7-68 


525i58 


23 


456739 


7-04 


981436 


-63 


475303 


7-67 


524697 


22 


39 


457162 


7-04 


981399 


-63 


475763 


]:n 


524237 


21 


40 


457584 


7-o3 


981361 


-63 


476223 


523777 


ao 


41 


9 -458006 


7-02 


9-981323 


•63 


9-476683 


7-65 


10-523317 


IQ 


42 


458427 
458848 


7-01 


981285 


•63 


477142 


7-65 


522858 18 1 


43 


7-01 


981247 


•63 


477601 


7-64 


522399 


\l 


44 


459268 


7-00 


981209 


•63 


478o5q 


7-63 


521941 


45 


459688 


6. 90 


981171 


•63 


478517 


7-63 


521483 


i5 


46 


460108 


6-98 


981133 


-64 


478975 


7-62 


521025 


14 


S 


460527 


6-98 


981095 


-64 


479432 


7-6i 


520568 


i3 


460946 
46 I 364 


6-97 


981037 


-64 


479889 


7.61 


520111 


la 


49 


6.96 


981019 


-64 


480345 


7.60 


519655 


ir 


5o 


461782 


6-95 


980981 


-64 


480801 


7-59 


519199 


10 


5i 


9 462199 


6-95 


9-980942 


• 64 


9-481257 


7-59 ■■ 


10-518743 


I 


52 


462616 


6-94 


980904 


-64 


481712 


7-58 1 


518288 


53 


463o32 


6-93 


980866 


-64 


482167 


7-57 


517833 


I 


54 


463448 


6.93 


980827 


-64 


482621 


7-57 


517379 
516925 


55 


463864 


6-92 


980789 


-64 


483073 


7-56 


5 


56 


464279 


6-91 


980750 


-64 


483529 


7^55 ! 


516471 


4 


U 


464694 


6-00 


980712 


-64 


483982 


7-55 ' 


5i6oi8 


3 


465 I 08 


6-90 


980673 


-64 


484435 


7-54 ! 


5i5565 


2 1 


59 


465522 


6.89 


980635 


-64 


484887 


7^53 1 


5i5ii3 


I 


60 


465935 


6-88 


980596 


-64 


485339 


7^53 1 


5i466i 


; 

.1 


•^., 


Cosiiie 


D. 


Sine 




Cotang. 


D. 1 


Tang. 1 M. j 








(73 


DEGR 


EK8.) 









SINS8 AND TANGENTS. (17 DEGREES.) 



36 



'^M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 


! O 
I 


9 -465035 
466348 


6-88 
6-88 


9-980596 

980558 


-64 

.64 


9-485339 

485791 


7-55 
7-52 


10.514661 
514209 
518758 


a 


466761 


6-87 


980519 


-65 


486242 


7.51 


3 


467173 

467535 


6-86 


980480 


.65 


486698 


7-5i 


5i33o7 


57 


4 


6-85 


980442 


• 65 


487148 


7-5o 


512857 ' 56 


5 


467996 
468407 


6-85 


980403 


-65 


487598 


7-49 


512407 55 


6 


6-84 


980364 


-65 


488043 


7-49 


5iio57 


54 


I 


468817 


6-83 


980325 


-65 


488492 


7-48 


5ii5o8 


53 


469227 


6-83 


980286 


-65 


488941 


7-47 


5iio;)9 


52 


9 


469634 
470040 


6-82 


980247 


-65 


489890 


7-47 


5io6io 


5i 


lO 


6-81 


980208 


• 65 


489888 


7-46 


510163 


5o 


II 


9.470455 


6-80 


9-980169 


-65 


9-490286 


7-46 


10-509714 


49 

48 


12 


470863 


6-80 


980130 


• 65 


490733 


7-45 


509267 


i3 


47 1 271 


6-70 
6.78 


980091 
980002 


-65 


491 180 


7-44 


508820 


S 


14 


471679 


-65 


491627 


7-44 


508878 


i5 


472086 


6.78 


98001 2 


-65 


492078 


7-43 


507927 


45 


i6 


472492 


6-76 


979973 


-65 


492519 


7-43 


507481 


44 


\l 


472898 


979034 


-66 


492965 


7.42 


507085 


43 


473304 


6.76 


979895 


-66 


498410 


7-41 


506590 


42 


»9 


473710 


6-75 


979855 


-66 


498854 


7.40 


506146 


41 


20 


4741 I 5 


6-74 


979816 


-66 


494299 


7-40 


5o570i 


40 


21 


9'4745iQ 
474Q23 
475327 


6-74 


9.979776 


-66 


9-494748 


7.40 


io-5o5257 


39 


22 


6-73 


979737 
979608 


-66 


495186 


7-89 


504814 


38 


23 


6-72 


-66 


495680 


7-88 


504870 


37 


i »4 


475730 


6-72 


.66 


496073 


7-37 


508927 


36 


j 25 


476133 


6.71 


979618 


.66 


49651 5 


7-37 


5o3485 


35 


26 


476536 


6-70 


979579 


.66 


496957 


7.36 


5o8o43 


34 


11 


476938 
477^40 


6-69 


979539 


-66 


497^99 


7-36 


502601 


38 


6.69 


979499 
979459 


-66 


497841 
498282 


7-35 


5o2i59 


32 


^9 


477741 
478142 


6-68 


-66 


7-34 


501718 


3i 


3o 


6-67 


979420 


-66 


498722 


7-34 


501278 


3o 


3i 


9-478542 


6.67 


9-979380 


-66 


9-499168 


7-33 


10 .500887 


It 


32 


478042 
479^42 


6.66 


979340 


-66 


499608 


7.38 


500897 
49995S 
499519 


33 


6-65 


979800 


-67 


5ooo42 


7-82 


U 


34 


479741 
480140 


6-65 


979260 


-67 


5oo48i 


7-81 


35 


6-64 


979220 


-67 


5oOQ20 

5oij59 


7-81 


499080 


23 


36 


480539 


6-63 


979180 


-67 


7-30 


498641 


24 


ll 


480937 
481334 


6.63 
6-62 


979140 
979100 


1^ 
.67 


501797 

5o22J5 


7-30 
7.20 


498208 
497765 
497828 


33 
23 


39 


481731 


6.61 


979059 


-67 


502672 


7.28 


31 


4o 


482128 


6-61 


979019 


-67 


5o8io9 


7-28 


496891 


20 


4i 


9-482525 


6-60 


9.978979 


-67 


9-5o3546 


7.27 


10-496454 


\t 


42 


482921 
4833 1 6 


6-59 


978989 


-67 


508982 


7-27 


496018 


43 


6.59 


978898 


.67 


5o44i8 


7-26 


495582 


\l 


44 


483712 


6-58 


978858 


.67 


5o4854 


7-25 


495146 


45 


484107 


6-57 


978817 


-67 


505289 


7-25 


4947 1 1 


i5 


46 


484501 


6-57 
6-56 


978777 
9787^6 


.67 


505724 


7-24 


494276 


14 


s 


484895 


t 


5o6i59 
506598 


7-24 


498841 


i3 


485289 


6-55 


978696 
978655 


7.28 


493407 12 1 


49 


485682 


6-55 


• 68 


507027 


7-22 


49297^ 


II 


5o 


48607^^ 


6-54 


978615 


-68 


507460 


7-22 


492040 


ro 


5i 


9.486467 


6-53 


9-978574 


-68 


9-507898 


7-21 


10-492107 


I 


52 


486860 


6-53 


978533 


-68 


5o8326 


7-21 


491674 


53 


487251 


6-52 


978408 


.68 


508759 


7-20 


491241 


I 


54 


487643 
488-)34 


6.5i 


978452 


.68 


509191 


7.19 


490809 


55 


6-5i 


978411 


.68 


509622 


?::§ 


400878 


5 


56 


488424 


6-5o 


978870 


• 68 


5ioo54 


489946 


4 


U 


488814 


6-5o 


978829 
978288 


68 


5 I 0485 


7-18 


489O15 


3 


489204 


6-49 


.68 


5iooi6 


7-17 


489084 


3 


59 


489593 
4B9982 


6-48 


978247 


.68 


5ii346 


7-i6 


488654 


I 


66 


6-48 


978206 


-68 


511776 


7.16 


488224 







Cosine 


D. 


Sine 


D. 


Cotang. 


_ D, 


Tang._ 


M. 



17 



(72 DEGRluKS.) 



36 



(18 DEGREES.) A TABLE OF LOGARITHMIC 



M. 


Sine 


D. 


Cosine ' 

i 


D. 


Tang. 


D. 


Cotacg. 


1 

60 





9-489082 
4903] I 


6-48 


9-978206 1 


.68 


9-511776 


7-16 


10-488224 


I 


6.48 


978165 1 


•68 


512206 


7.16 


487794 


5q 


a 


490709 


6.47 
6-46 


978124 ' 


• 68 


512635 


7.15 


487365 


5tl 


3 


491147 


978083 ' 


•69 


5i3o64 


7-14 


486936 


57 


4 


491535 


6-46 


978042 1 


•^ 


513493 


7-14 


4865o7 


56 


5 


491922 
492308 


6-45 


978001 


•69 


oi3q2i 
514349 


7-13 


486079 


55 


6 


6-44 


977900 


.69 


7-i3 


485651 


54 


I 


492605 
493081 


6-44 


977018 ; 


.69 


514777 


7.12 


485223 


53 


6-43 


977835 


.69 


510204 


7-12 


484796 


5a 


9 


493466 


6-42 


.69 


5i563i 


7-11 


484369 
483943 


5i 


10 


49385i 


6-42 


977794 


.69 


516057 


7-10 


5o 


II 


9-494236 


6.41 


9.977752 


.69 


9.016484 


7.10 


10.483576 


49 


12 


494621 


6.41 


9777" 


.69 


016910 


7-09 


483090 


48 


i3 


495oo5 


6-40 


977660 
977628 


.69 


517335 


?:S 


482665 


47 


14 


493388 


6.39 


.69 


517761 


482239 
481815 


46 


i5 


495772 
49604 


6.39 


977586 


.69 


5i8i85 


7.08 


45 


i6 


6.38 


977544 


.70 


518610 


?:S 


481390 


44 


17 


496537 


6-37 


9775o3 


.70 


519034 


480066 
480542 


43 


i8 


496019 
497301 


6.37 


977461 


.70 


519458 


7.06 


42 


19 


6-36 


977419 


.70 


519882 


7-05 


480118 


41 


ao 


497683 


6-36 


977377 


.70 


52o3o5 


7.05 


479695 


40 


31 


9.498064 


6.35 


9-977335 


.70 


9.520728 


7.04 


10.479272 


It 


2a 


498444 


6.34 


977293 


.70 


52ii5i 


7 -03 


478849 


23 


498825 


6-34 


977231 


• 70 


521573 


7.03 


478427 
478000 


37 


U 


499204 


6.33 


977209 


• 70 


521995 


7.03 


36 


25 


499584 


6-32 


977167 


• 70 


522417 


7.02 


477583 


35 


26 


499963 


6-32 


977120 


.70 


522838 


7.02 


477162 


34 


11 


5oo342 
500721 


6.3i 
6-31 


977083 
977041 


.70 
.70 


523259 
523680 


7-01 
7.01 


476741 
476320 


33 

33 


1? 


501090 
001476 


6.3o 
6.29 


976999 
976907 


.70 
.70 


024100 
524520 


7.00 
6.99 


475900 
475480 


3i 
3o 


3i 


9-5oi854 


6.29 


9.976014 
97683o 


.70 


9.524939 


t:^ 


10.475061 


'9 i 


32 


5o223l 


6.28 


•71 


52535q 


474641 


28 


33 


502607 


6-28 


.71 


520778 


6.98 


474222 


11 


34 


502984 
5o336o 


6-27 


976787 
976745 


.71 


526197 


6.97 


4738o3 


35 


6.26 


•71 


5266i5 


?;^ 


473385 


25 


36 


5o3735 


6.26 


976702 


•71 


527033 


472967 


24 


U 


5o4iio 


6.25 


976660 


•71 


527451 


6.96 


472049 


a3 


5o4485 


6.25 


976617 


•71 


527868 
528285 


6.95 


472132 


22 


39 


504860 


6-24 


976574 


.71 


6.95 


471715 


21 


40 


5o5234 


6.23 


976532 


.71 


528702 


6.94 


471298 


20 


41 


9 -505608 


6-23 


9.976489 


.71 


9-529119 


6.93 


10.470881 


^2 


42 


505981 


6-22 


976446 


•71 


029530 


6.93 


470465 


18 


43 


5o6354 


6.22 


976404 


.71 


529900 
53o366 


6.93 


47oo5o 


]l 


44 


506727 


6.21 


976361 1 


•71 


6.92 


469634 


45 


507099 


6.20 


676318 ' 


•71 


530781 


6.91 


460219 


i5 , 


46 


5oi47i 


6-20 


976275 1 


.71 


531196 


6.91 


468804 


14 


S 


507843 
5o82i4 


6.19 
6.19 


976232 i 
976189 : 


.72 
.72 


53i6ii 
532025 


6.90 

6. go 

6.^ 


468389 
467075 
467061 


i3 
13 


49 


5o8585 


6.18 


976146 1 


.72 


532439 
532853 


u i 


5o 


508956 


6.18 


976103 


.72 


6.89 


467147 


LOJ 


5i 


9.509326 


6.17 

6-i6 


9.976060 


.72 


9-533266 


6.88 


10.466734 
466321 


l\ 


52 


509696 


976017 


.72 


533679 


6.88 


53 


5ioo65 


6.16 


975974 


• 72 


534092 


6.87 


465908 


h 


54 


5x0434 


6-i5 


975o3o , 
975887 


.72 


534504 


6.87 


465496 
465084 


55 


5io8o3 


6.i5 


.72 


534916 
535328 


6-86 


5 


56 


511172 


6-14 


975844 


•72 


6-86 


464672 
464261 


4 


n 


5ii54o 


6-i3 


975800 


.72 


535739 


6-85 


3 


1\IV^ 


6-i3 


975757 


.72 


536 i5o 


6-85 


463850 


3 


59 


6-12 


975714 
970670 


•72 


536561 


6-84 


463439 
463028 


I 


66 


512642 


6-12 


.72 


536972 


6-84 







Cosine 


D. 


Sine ! 


D. 


Co tang. 


D. 


lang. 



(71 D£(^R££6.) 



SINES AND TANGENTS. (19 DEGREES.) 



37 



M." 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


CotAn^. 







9-512642 


6-12 


9-975670 


.73 


9-536972 
537882 


6.84 


10.4680:8 


60 


I 


5 I 3009 
513375 


t 


-II 


975627 


.73 


6 


83 


462618 


U 


a 


6 


■11 


975583 


.73 


537792 
588202 


6 


-83 


462208 


3 


5i374i 


6 


• 10 


975539 


"^i 


6 


.82 


461708 
461889 


57 


6 


514107 


6 


09 


975496 
975452 


"^l 


58861 1 


6 


82 


56 


5 


514472 


6 


09 


-73 


589020 


6 


81 


460980 
460371 


55 


6 


514837 


6 


• 08 


975408 


•73 


589429 


6 


81 


54 


I 


5l5202 


6 


.08 


975365 


•7? 


589887 
540245 


6 


.80 


460163 


53 


5i5566 


6 


.07 


975321 


-73 


6 


.80 


459755 


52 


9 


5 I 5980 


6 


:S 


975277 
975233 


•73 


540658 


6 


■79 


459347 
458989 


5i 


to 


516294 


6 


.73 


541061 


6 


•79 


5o 


II 


9.516657 


6 


o5 


9'975i89 
975145 


•73 


9.541468 


6 


78 


10-458532 


^ 


la 


517020 


6 


o5 


•73 


541875 
542281 


6 


.78 


458125 


i3 


517382 


6 


04 


975101 


•7? 


6 


77 


457719 


ii 


14 


517745 


6 


04 


975057 
975oi3 


.73 


542688 


6 


U 


457812 


i5 


518107 


6 


o3 


.73 


548094 


6 


406906 


45 


i6 


518468 


6 


o3 


974969 


•74 


543499 


6 


76 


456301 


44 


\l 


518829 


6 


02 


974925 


•74 


548905 


6 


75 


456095 


43 


519190 
5i95di 


6 


01 


974880 


•74 


544810 


6 


75 


4556Q0 


42 


19 


6 


01 


974836 


•74 


544715 


6 


74 


455285 


41 


20 


519911 


6 


00 


974792 


•74 


545119 


6 


74 


454881 


40 


21 


9.520271 


6 


00 


9.974748 


•74 


9 545524 


6 


73 


10.454476 


39 


22 


52o63i 


5 


99 


974703 


•74 


545928 


6 


73 


454072 


38 


23 


520990 


5 


99 


974659 


•74 


546881 


6 


72 


458669 


U 


34 


521349 


5 


98 


974614 


•74 


546785 


6 


72 


458265 


a5 


521707 


5 


98 


974570 


•74 


547188 


6 


71 


452862 


35 


a6 


522066 


5 


97 


974525 


•74 


547540 


6 


71 


452460 


34 


11 


522424 


5 


96 


974481 


•74 


547948 
548845 


6 


70 


452057 


33 


522781 


5 


96 


974436 


•74 


6 


ll 


45 1 655 


32 


29 


523 i38 


5 


95 


974391 


•74 


548747 


6 


451253 


3i 


3o 


523495 


5 


95 


974347 


•75 


549149 


6 


69 


45o85i 


3o 


3i 


9-523852 


5 


94 


9-974302 


"'^ 


9.549550 


6 


68 


io.45o45o 


29 


32 


524208 


5 


94 


974257 


•7f 


549951 
55o352 


6 


68 


450049 


28 


33 


524564 


5 


93 


974212 


•75 


6 


67 


449648 


27 


34 


524920 


5 


93 


974167 


•75 


550752 


6 


U 


449248 


a6 


35 


525275 

52563o 


5 


92 


974122 


.75 


55ii52 


6. 


448848 


25 


36 


5 


91 


974077 


•75 


55i552 


6- 


66 


448448 


24 


ll 


525q84 
526339 
526693 


5. 


91 


974082 


•75 


551952 


6- 


65 


448048 


23 


5 


90 


973987 


"^i 


552851 


6 


65 


447649 


32 


39 


5 


90 


973942 


.75 


552750 


6- 


65 


447250 


ai 


40 


527046 


5. 


89 


978897 


.75 


553 149 


6. 


64 


446851 


20 


41 


9-527400 


5 


89 


9.978852 


"^i 


9.553548 


6. 


64 


10-446452 


J? 


4a 


527753 
528105 


5. 


88 


978807 


.75 


558946 


6- 


63 


446054 


43 


5 


88 


978761 


-75 


554344 


6- 


63 


445656 


17 


44 


528458 


5 


87 


978716 


.76 


554741 


6- 


62 


445259 16 1 


45 


528810 


5 


87 


978671 


.76 


555189 


6- 


62 


444861 


i5 


46 


529161 


5 


86 


978625 


-76 


555586 


6- 


6i 


444464 


14 


4*2 


5295 I 3 


5 


86 


973580 


-76 


555988 
556320 


6. 


61 


4/14067 


i3 


529864 


5 


85 


978585 


-76 


6 


60 


448671 


12 


49 


53021 5 


5 


85 


978489 


-76 


556725 


6 


60 


448275 


II 


5o 


53o565 


5 


84 


973444 


-76 


557121 


6 


59 


442879 


10 


5i 


9 53091 5 


5 


84 


9-973898 


-76 


9-557517 


6 


59 


10.442488 


I 


53 


531265 


5 


83 


978852 


.76 


557913 

5583o8 


6. 


59 


442087 


53 


53i6i4 


5 


82 


978807 


.76 


6 


58 


441692 


I 


54 


53io63 

532JI2 


5 


82 


978261 


.76 


558702 


6 


58 


441298 


55 


5 


81 


978215 


.76 


559097 


6 


57 


440903 


5 


56 


532661 


5 


81 


978169 


.76 


559491 


6. 


U 


440300 
4401 i5 


4 


U 


533009 


5 


80 


978124 


'76 


559885 


6- 


3 


533357 


5 


80 


978078 
978033 


-76 


560270 
560678 
56io66 


6. 


56 


489721 


2 


59 


533704 


5 


79 


•77 


6. 


55 


480827 
438934 


1 


60 


534052 


5.78 


972986 


•77 


6.55 



M. 


L-, 


Cosine 


E 


. 


Sine i 


D. 


Cotan^. 


D. 


Tanir 



(70 DEORSSS.^ 



88 



(20 DEGREE8J A TABLE OF LOGARITHMIC 



M. 


Sine 


D. 


Cosine | 

1 


D. 


Tang. 


D. 


Cotang. 


( 





9 -534052 


5.78 


9.972986 1 


•77 


9-56io66 


6.55 


10-438034 


^ 


I 


53439^ 


5-77 


972040 1 


•77 


561439 


6.54 


438541 i 59 


a 


534743 


5-77 


972894 


•77 


56i85i 


6-54 


^38149 


58 


3 


535oo2 
536438 


5-77 


972848 


•77 


562244 


6.53 


437756 


u 


4 


5-76 


972802 


•77 


562636 


6.53 


437364 


5 


535783 


5.76 


972755 


'77 


563023 


6.53 


436972 


55 


6 


536i2o 


5-75 


972709 


'77 


563419 


6-52 


43608 I 


54 


2 


536474 


5-74 


972663 ' 


•77 


5638 1 1 


6-52 


436180 
435798 


53 


5368 1 8 


5-74 


972617 


•77 


564202 


6-51 


52 


9 


537163 


5.73 


972070 


•77 


564002 
564983 


6-51 


435408 


5i 


lO 


537507 


5.73 


972524 


•77 


6.5o 


435017 


5o 


II 


9-537851 


'5-72 


9-972478 , 


•71 


9-565373 


6.5o 


10-434627 


S 


12 


538194 


5-72 


972431 


.78 


565763 


6.49 


434237 


i3 


538538 


5.71 


972385 


.78 


566 1 53 


6-49 


433847 


47 


14 


538880 


5-71 


972338 , 


•'?? 


566542 


5.40 


433408 


46 


i5 


53^223 


5-70 


972291 , 


"'? 


566932 


6-48 


433068 


45 


i6 


539565 


5-70 


972245 ■ 


•''? 


567320 


6-48 


432680 


44 


17 


539907 


5-69 


972198 [ 


•''o 


567709 


6-47 


432291 


43 


i8 


540249 


5-6q 


9721D1 


.78 


568oq8 


6-47 


431902 


42 


19 


540590 


5-68 


972105 1 


.78 


5684^6 


6.46 


43ioi4 


41 


20 


540931 


5-68 


972058 j 


.78 


568873 


6-46 


431127 


40 


31 


9-541272 


5-67 


9-9720x1 


.78 


9.569261 


6.45 


10-430739 


39 


97 


54i6i3 


5.67 


971964 


•7? 


569648 


6-45 


43o3o2 


38 


33 


541953 


5-66 


971917 


•■'q 


570035 


6.45 


429965 


37 


24 


542 2o3 
542632 


5-66 


971870 


"^^0 


570422 


6-44 


429078 


36 


i5 


5-65 


971823 ' 


•7? 


570809 


6-44 


429191 


35 


36 


542971 ' 


5-65 


971776 ' 


.78 


571190 


6-43 


428805 


34 


11 


543310 


5-64 


971729 i 


•79 


571081 


6-43 


428419 


33 


543649 


5-64 


971682 • 


•79 


571967 


6-42 


428033 


32 


29 


543987 , 


5-63 


971635 


•79 


572352 


6-42 


427648 


3i 


So 


544325 I 


5-63 


971588 


•79 


572738 


6.42 


427262 


3o 


3i 


9-544663 1 


5-62 


9-971540 


•79 


9.573123 


6.41 


10-426877 


29 


34 


540000 , 


5-62 


971493 ; 


•79 


573507 


6.41 


426493 


28 


33 


545338 


5-6i 


971446 


•79 


573892 


6-40 


426108 


U 


34 


545674 


5.61 


971398 


•79 


574276 


6-40 


425724 


35 


54601 1 


5.60 


97i3di 


•79 


574660 


6-39 


420340 


25 


36 


546347 


5.60 


97i3o3 


•79 


070044 


6-39 


424906 


24 


U 


546683 


5-39 


971256 


•79 


575427 


6.39 


424073 


23 


547019 


5-50 


971208 < 


•79 


570610 


6-38 


424190 


23 


39 


547354 


5-58 


971161 ' 


•79 


576193 


6.38 


423807 


21 


40 


547689 


5-58 . 


971113 ; 


•79 


576576 


6.37 1 


423424 


20 


41 


9.548024 


5-57 


9-971066 1 


.80 


9.576958 


6.37 


10.423041 


10 


47 


548359 


5«57 


971018 i 


.80 


577341 


6.36 i 


422659 


id 


43 


548693 


5-56 


970970 ', 


.80 


577723 


6.36 


422277 


11 


44 


549027 


5-56 


970922 


-80 


578104 ■ 


6-36 


421896 


16 


45 


549360 


5-55 


970874 


.80 


078486 ■ 


6-35 


42i5i4 


A 


46 


549693 


5-55 


970827 


.80 


578S67 ; 


6.35 


421 i33 


14 


% 


550026 


5-54 


970779 


-80 


579248 j 


6.34 


420752 


i3 


55o359 


5-54 


970731 


-80 


579629 ! 


6.34 


420371 


13 


49 


550692 


5-53 


970683 : 


.80 , 


580009 ! 


6.34 


419991 


II 


66 


55io24 


5-53 


970635 ; 


.80 


58o389 


6.33 ! 


419611 


10 


5i 


g-55i356 


5-52 


9-970586 i 


.80 ' 


9-080769 : 


6.33 


10-419231 


t 


53 


55i68t 
552018 


5-52 


970538 ' 


-80 , 


581149 1 


6.32 


4i885i 


53 


5-52 


970490 


.80 


58i528 ! 


6-32 


418472 1 7 


54 


552349 


5-51 


970442 


.80 


581907 j 


6-32 


418093 


6 


55 


552680 


5-5i 


970394 


.80 


082286 


6-31 


417714 


5 


56 


553oio 


5-5o 


970345 1 


.81 


582665 


6-31 


417335 


4 


58 


553341 


5-5o 


970297 


-81 


583043 


6-30 


416957 
416578 


3 


553670 


5-49 


970249 


• 81 


583422 i 


6-30 


3 


59 


554000 1 


5.49 


970200 ' 


.81 


583800 1 


6.29 


416200 


I 


66 

■ 


554329 \ 


5-48 


970.52 : 


.81 : 


584177 ; 


6-29 


415823 


• ^^^» 


Cosine ', 


D. 


Sine 


D. . 


Cotang. t 


i>- : 


Tan«. _ 


M. 



(69 DEQRSSS.) 



SINKS AUD TANGENTS. (21 DEGREES.) 



39 



M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 


o 


0' 554329 
554658 


5.48 


9-970152 


-81 


9 •584177 
584555 


6-29 


10-4x5823 


1 


5.48 


970103 


.81 


6 


11 


4x5445 


^ 


2 


554987 


5-47 


970055 


-81 


584Q32 
585^09 


6- 


4i5o68 


58 


3 


5553x5 


5-47 
5-46 


970006 
969957 


-81 


6 


28 


414691 


57 


4 


555643 


-81 


585686 


6 


27 


414314 


56 


5 

6 


555971 
556299 
556626 


5.46 
5.45 


969S60 


.81 
-81 


586062 
586439 


6 
6. 


27 
27 


413038 
4i356i 


55 
54 


i 


5.45 


96981 1 


•8x 


5868x5 


6- 


26 


4i3i85 


53 


556953 


5-44 


969762 


-8x 


587190 


6 


26 


412810 


52 


9 


557280 


5.44 


969^65 


-8x 


587566 


6 


25 


412434 


5i 


lO 


557606 


5.43 


-8x 


587941 


6 


25 


412059 


5o 


II 


Q. 557932 

558258 


5-43 


9-969616 


.82 


9-5883x6 


6- 


25 


10 41 1684 


g 


!2 


5-43 


969567 
969518 


-82 


588691 


6- 


24 


41 1 309 


i3 


558583 


5-42 


-82 


589066 


6- 


24 


4xoo34 
4io56o 


47 


14 


558909 


5.42 


969469 


-82 


589440 


6- 


23 


46 


i5 


559234 


5.41 


969420 


-82 


589814 


6- 


23 


410186 


45 


i6 


550558 


5-41 


969370 


-82 


590188 


6- 


23 


409812 


44 


\l 


559883 


5.40 


969321 


-82 


590562 


6- 


22 


409438 


43 


560207 


5.40 


969272 


-82 


590035 
591^08 


6- 


22 


409065 


42 


19 


56o53i 


5.39 


969223 


-82 


6- 


22 


408692 


41 


ao 


56o855 


5.39 


969173 


-82 


59X68X 


6 


21 


408319 


40 


21 


9-561178 


5-38 


9-969x24 


.82 


9 592054 


6- 


21 


XO -407946 
407574 


39 


22 


56i5oi 


5.38 


969075 


.82 


592426 


6. 


20 


38 


23 


561824 


5.37 


960025 
968976 


.82 


592798 


6 


20 


407202 


U 


S4 


56s 146 


5.37 
5.36 


.82 


593170 


6 


19 


406829 


23 


562468 


968926 
968877 


• 83 


593542 


6 


XO 

x8 


406458 


35 


26 


562790 


5.36 


.83 


5939x4 


6 


406086 


34 


11 


563II2 


5.36 


968827 


.83 


594285 


6 


x8 


405715 
405344 


33 


563433 


5.35 


968777 
968728 


.83 


594656 


6 


18 


32 


19 


563755 


5-35 


-83 


595027 
595398 


6 


»7 


404973 


3i 


3o 


564075 


5.34 


968678 


-83 


6 


17 


404602 


So 


3i 


9-564396 


5.34 


9.968628 


-83 


9-595768 


6 


\l 


10.404232 


29 


32 


564716 


5-33 


968578 


-83 


596138 


6 


4o3862 


28 


33 


565o36 


5.33 


968528 


-83 


596508 


6 


16 


403492 


ll 


34 


565356 


5.32 


968479 


-83 


596878 


6 


x6 


4o3i22 


35 


565676 


5-3a 


968429 


-83 


597247 


6 


x5 


402753 


25 


36 


565995 
5663 14 


5-3i 


968379 


.83 


5976x6 


6 


i5 


402384 


24 


ll 


5-3i 


968329 
968278 


.83 


597085 
598 J 54 


6 


i5 


40201 5 


23 


566632 


5-3i 


.83 


6 


14 


401646 


22 


39 


566951 


5-3o 


968228 


-84 


598722 


6 


14 


401278 


21 


40 


567269 


5.30 


968x78 


.84 


599091 


6 


x3 


400909 


20 


41 


9.567587 


5-29 


9-968128 


.84 


9-599459 


6 


i3 


10 -400541 


\t 


42 


567904 
568222 


5-29 
5-28 


968078 


.84 


599827 


6 


x3 


400173 


43 


968027 


.84 


600194 


6 


xs 


399806 


17 


44 


568539 


5.28 


967977 


.84 


6oo562 


6 


X2 


399438 . 16 1 


45 


568856 


5.28 


967027 
967876 


-84 


600929 


6 


IX 


39007 X 


.15 


46 


569172 


5-27 


.84 


601296 


6 


XI 


398704 


14 


fi 


569488 


5-27 
5-26 


967826 


.84 


60x662 


6 


XI 


398338 


iJ 


569804 


967775 


.84 


602029 
602395 


6 


XO 


397971 


12 


^ 


570120 


5-26 


967725 


.84 


6 


10 


397605 


II 


5o 


570435 


5-25 


967674 


.84 


60270X 


6 


XO 


397239 


10 


5i 


9.570751 


5-25 


9-967624 


.84 


9-6o3i27 
603490 
6o3858 


6 


09 


10-396873 


I 


5a 


571066 


5-24 


967573 


-84 


6 


09 


396507 


53 


571380 


5.24 


967522 


.85 


6 


.s 


396142 


I 


54 


571695 


5.23 


967471 


.85 


604223 


6 


395777 


55 


572009 
572323 
572636 


5-23 


967421 


.85 


604588 


6 


■08 


395412 


5 


56 


5-23 
5.22 


967370 
967319 
967268 


-85 

-85 


6o4q53 
6o5Jn 


6 
6 


•07 
.07 


395047 
394683 


4 
3 


572950 


5«22 


-85 


6o5682 


1 6 


•07 


394318 


2 


^ 


573263 


5-21 


967217 


-85 


606046 


6 


•06 


393954 
393590 


I 


60 


573575 


5-21 


967166 


-85 


6064x0 


6-06 





L , 


Gosino 


D. 


Sine 


D. 


Cotang. 


D. 


Tang. 



(08 DEORSSS.^ 



iO 



^22 DB3^REES.) A TABLE OF LOGARITHMIC 



m:. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


1 
60 





9"573575 
573868 


5-31 


9-967166 


.85 


9-606410 


6.06 


10-398590 


I 


5.30 


9671 i5 


.85 


606778 


6.06 


39822- 
892860 


5? 


3 


574200 


5.30 


967064 


.85 


607187 


6.o5 


3 


574512 


5-19 


967013 


.85 


607500 


6.o5 


892500 


u 


4 


574834 


5-19 


966961 


-85 


607863 


6 04 


392187 
891775 


5 


575 I 36 


5-19 
5-i8 


966910 


• 85 


608225 


6.04 


55 1 


6 


5?5^J 


966859 


• 85 


6o8588 


6.04 


891412 


54 


I 


5.18 


966808 


.85 


6o8o5o 
609^12 


6-o3 


391050 i 53 1 


576069 


5-17 


966756 


.86 


6-03 


890688 


53 


9 


576379 
576669 


5-17 
5-16 


966705 


.86 


609674 


6 o3 


800826 


5i 


10 


966653 


-86 


610086 


6-02 


389964 


5o 


II 


9.576099 
577300 
577618 


5.16 


9-966603 


-86 


9.610807 


6-02 


10 889608 


^ 


13 


5.16 


966550 


-86 


610739 


6.02 


889241 


i3 


5.i5 


966499 


•86 


611120 


6-01 


388880 


ii 


14 


577937 
578336 


5.i5 


966447 
966895 


.86 


61 1480 


6-01 


388520 


i5 


5.14 


.86 


611841 


6.01 


888159 


45 


i6 


578545 


5.14 


966844 


.86 


612201 


6.00 


887709 
887489 


44 


\l 


578853 


5-i3 


966292 


-86 


6i256i 


6.00 


43 


579162 


5-13 


966240 


• 86 


612921 


6-00 


887079 
886719 
386359 


43 


«9 


579470 


5.i3 


966188 


.86 


618281 


5.99 


41 


20 


579777 


5.13 


966186 


• 86 


618641 


5-99 


40 


31 


9 -580085 


5-13 


9-966085 


•87 


9.614000 


5-98 


10-886000 


^ 


23 


580892 


5. II 


966088 


.87 


6i485q 
614718 


5.98 


385641 


23 


580699 
58roo5 


5. II 


965981 


•87 


5.98 


385282 


U 


24 


5. II 


965928 
965876 


-87 


61 5077 
6i5485 


5-97 


884928 
884565 


25 


58i3i2 


5-10 


•87 


5-97 


35 


26 


58i6i8 


5-10 


965824 


-87 


615798 
6i6i5i 


i:^ 


884207 


34 


U 


58iQ24 


5.09 


965772 


•87 


888849 


33 


582229 
582535 


5-09 


965720 


•87 


6i65o9 


5-96 


888491 
388 1 33 


33 


?9 


5.09 
5.08 


965668 


•87 


616867 


5-96 


3i 


3o 


582840 


9656 1 5 


•87 


617224 


5.95 


882776 


3o 


3i 


9-583145 


5.08 


9.965568 


.87 


9.617582 


5.95 


10.382418 


^ 


32 


583449 


5.07 


96551 1 


•87 


611989 
618295 


5-95 


882061 


33 


583754 


5.07 
5.06 


965458 


.87 


5.94 


881705 


U 


34 


584058 


965406 


ii 


6i8652 


5.94 


881848 


35 


584361 


5.06 


965853 


619008 


5-94 


380992 
380686 


35 


36 


584665 


5.06 


965801 


.88 


619864 


5.98 


34 


ll 


584968 


5.o5 


965248 


.88 


619721 


5.93 


880279 


33 


585272 


5.o5 


965195 


.88 


620076 
620482 


5.98 


879924 
879568 


33 


J9 


5g5574 


5-04 


965148 


.88 


5.92 


31 


40 


585877 


5-o4 


965090 


.88 


620787 


5.92 


879218 


30 


41 


9- 586179 
586462 


5-03 


9.965087 


.88 


9.621142 


5.92 


10-378858 


:? 


42 


5.o3 


964984 


.88 


621497 
621852 


5.91 


3785o3 


43 


586783 


5-o3 


96498 1 


.88 


5.91 


378148 


17 


44 


587085 


5-03 


964879 


.88 


622207 


5.90 


377793 
877489 
377085 
376781 
376877 


16 


45 


587386 


5-03 


964826 


.88 


622561 


5.90 


i5 


46 


587688 


5-01 


964773 


.88 


622915 


5.90 
5.89 
5.89 


i4 


S 


587989 
588289 


5-01 
5-01 


964719 
964066 


.88 
.89 


628269 
628628 


•3 

f2 


49 


588590 


5-00 


9646x8 


.89 


628076 
624880 


5.89 
5.88 


876024 


II 


5o 


588890 


5-00 


964560 


.89 


875670 


P 


5i 


9 589100 
589489 


4-99 


9.964507 


.89 


9.624688 


5-88 


10-875317 


t 


52 


4-99 


964454 


.89 


625086 


5.88 


874964 
374012 


53 


589780 
590088 


4-99 
4.98 


964400 


.89 


625888 


5.87 


I 


54 


964847 


.89 


625741 


5.87 


374259 


55 
56 


590887 
590686 


4.98 
4-97 


964294 
964240 


.89 
.89 


626093 
626445 


5.87 
5.86 


878907 
373555 


5 
4 


U 


590984 


4-97 


964187 
964183 


.89 


626797 


5.86 


378208 


3 


591282 


4-97 
4-96 


.89 


627149 


5.86 


372851 


9 


59 


591580 


964080 


.89 


627501 


5.85 


372499 
372140 


I 


6o 


591878 


4-0 


964026 


-89 


627852 


5-85 







Cofline 


D. 


Sine 


D. 


Cotniig. 


D. 


Tang. J 


M. 



(67 DEORSXS.) 





SINES AND TANGENTS 


. (23 EEGRBKS.) 


41 


M. 


Sine 


D. 


Cosine 


D. ! 


Tang. 


D. 


Cotang. 







9.591878 ' 


4.96 


9.964026 


.89 : 


9-627852 1 


5.85 


10.372148 


60 


I 


592176 


4.95 


968972 


.89 


628208 i 


5.85 


371797 


u 


a 


592473 


4.95 


968919 


.89 


628554 


5-85 


871446 


3 


592770 


4-95 1 


963865 


■90 


628905 


5.84 


871095 


u 


4 ! 


598067 
593363 


4-94 ' 


96881 1 


90 1 


629255 


5.84 


870745 


5 


4.94 


968757 


.90 


629606 


5.83 


870394 


55 


6 


593659 
598955 


4.98 


968704 


•90 


629956 
680806 


5.83 


870044 


54 


I 


4.98 


968650 


.90 j 


5-83 


369694 


53 


594251 


4.93 


968596 


.90 


63o656 


5.83 


869844 


5a 


9 


594547 


4-92 


963542 


.90 


63ioo5 


5.82 


860995 


5i 


IC 


594842 


4.92 


968488 


.90 


63i355 


5.82 


368645 


5o 


II 


9 595137 


4-91 


9.968484 


.90 


9.681704 


5-82 


10.368296 


§ 


12 


595432 


4-91 


968879 


.90 


682o58 


5-8i 


367947 


i3 


595727 


4-91 


968825 


.90 


682401 


5-8i 


867399 
367260 


% 


F4 


596021 


4.90 


968271 


.90 


682750 


5.81 


i5 


5963 1 5 


4-00 


968217 


.90 


688098 


5-80 


366902 
366558 


45 


i6 


596609 
596903 


4.89 


968168 


.90 


683447 
688795 


5-8o 


44 


\l 


4-89 


968108 


.91 


5-8o 


3662o5 


43 


597196 


4.89 
4-88 


968054 


.91 


684148 


5-79 


365857 


42 


19 


597400 


962999 


.91 


684490 


5-79 


365510 


41 


20 


597783 


4-88 


962945 


.91 


684888 


5.79 


365i62 


40 


21 


9.598075 


4-87 


9.962890 


•91 


9.685i85 


5-78 


10 .3648 1 5 


It 


22 


598868 


4-87 


962886 


.91 


635582 


5-78 


364468 


23 


598660 


4-87 


962781 


.91 


685879 


5-78 


3641 21 


ll 


24 


598952 


4-86 


962727 


.91 


686226 


5.77 


868774 


25 


599244 


4-86 


962672 


.91 


686572 


5-77 


868428 


35 


26 


599536 


4-85 


962617 


.91 


686919 


5-77 


368081 


34 


^ 


599827 
6001 18 


4-85 


962662 


.91 


687265 


5-77 


362785 


33 


4-85 


962508 


.91 


687611 


5-76 


362889 


32 


29 


600409 


4.84 


962453 


.91 


687955 
638802 


5-76 


362044 


3i 


3o 


600700 


4-84 


962898 


.92 


5-76 


361698 


3o 


3i 


9.600900 
601280 


4-84 


9.962848 


.92 


9 '688647 


5.75 


io.36i353 


20 


32 


4-83 


962288 


.92 


688992 


5.75 


361008 


28 


33 


601570 


4-83 


962288 


.92 


689337 


5-75 


860668 


ll 


34 


601860 


4-82 


962178 


.92 


689682 


5-74 


360818 


35 


602 1 5o 


4-82 


962128 


.92 


640027 


5-74 


359978 


25 


36 


602489 


4-82 


962067 


.92 


640871 


5-74 


359629 


24 


U 


602728 


4-8i 


962012 


.92 


640716 


5.73 


359284 


23 


608017 
6o83o5 


4-8i 


961957 


.92 


641060 


5.73 


358940 
358596 


22 


39 


4-8i 


961902 


.92 


641404 


5.73 


21 


40 


603594 


4- 80 


961846 


.92 


641747 


5.72 


358253 


20 


41 


9.608882 


4-8o 


9.961701 
961735 


.92 


9.642091 


5-72 


10.357909 
357566 


\l 


42 


604170 


4-79 


.92 


642484 


5.72 


43 


604457 
604745 


4-79 


961680 


•92 


642777 


5-72 


357228 


\l 


44 


4-79 


961624 


.93 


648120 


5.71 


356880 


45 
46 


6o5o32 
605819 


4-78 
4-78 


961569 
96i5iJ 


.93 
.93 


648468 
648806 


5.71 
5.71 


35653? 
356194 
855852 


it} 

14 


S 


6o56o6 


4-78 


961458 


.93 


644148 


5-70 


i3 


605892 


4-77 


961402 


.93 


644490 


5.70 


855510 


13 


19 


606179 


4-77 


961846 


.93 


644882 


5.70 


355i68 


II 


5o 


606465 


4-76 


961290 


•93 


645174 


5.69 


354826 


10 


5: 


9.606751 


4-76 


0-961235 


.93 


9.645516 


5.69 


K>. 354484 


I 


53 


607086 


1 4-76 


961 170 
961123 


.93 


645857 


5.69 


' 854143 


53 


607822 


1 4-75 


•9? 


646199 


5.69 


35880I 


I 


54 


607607 


1 4-75 


961067 


93 


646540 


5-68 


353460 


55 


607892 


4-74 


961011 


.93 


646881 


5-68 


353119 

, 352778 

352488 


5 


56 


608177 


4-74 


960955 


.93 


647222 


5-68 


4 


57 


608461 


4-74 


960899 
960843 


.93 


647562 


5-67 


3 


58 


608745 


1 4-73 


•94 


647908 
648243 


5-67 


352097 
351757 


2 


^ 


609029 
609813 


4-73 


960786 


•94 


5.67 


I 


60 


4-73 


960780 


•94 


648583 


5.66 

1 


351417 







Cosine 


! D. 


Sine 


1 D. 


Cotan^. 


1 D. 


• Tang. 


M. 








(66 


DSOl 


IBES.) 









12 



(84 DEGREES.) A TABLE OF LOGARITHMIC 



M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 


o 


9'6o93i3 


4-73 


9 • 960780 


•94 


9-648588 


5-66 


10.351417 


I 


609597 
609880 


4 


.72 • 


960674 


-94 


648928 


5 


66 


35107-7 

350737 
35c39S 
85oo58 


So 


i 


4 


■ 72 


960618 


.94 


649268 


5 


66 


5§ 


3 


610164 


4 


•72 


960561 


•94 


649602 


5 


66 


^ ' 


4 


610447 


4 


•71 


96o5o5 


.94 


649942 


5 


65 


5 


610729 


4 


•71 


960448 


•94 


65o28i 


5 


65 


349719 


55 1 


6 


611012 


4 


.70 


960892 


.94 


65o62o 


5 


65 


349880 


54 1 


I 


61 1 294 


4 


.70 


960835 


.94 


650959 


5 


64 


349041 


53 ; 


61 1 576 


4 


.70 


960279 


.94 


65i297 
65 I 636 


5 


64 


848703 ! 5a 


9 


6ii858 


4 


.6q 


960222 


.94 


5 


64 


348364 1 5i , 


10 


612140 


4 


69 


960165 


•94 


651974 


5 


63 


348026 


5o 


II 


9-612421 


4 


69 


0-960109 


-95 


9-652812 


5 


63 


10-847688 


it 


12 


612702 


4 


68 


960052 


.95 


65265o 


5 


63 


847350 


i3 


612983 


4 


68 


959995 


.95 


652988 
658826 


5 


68 


347012 


11 


14 


618264 


4 


.67 


959988 


-95 


5 


62 


346674 


i5 


613545 


4 


67 


959882 


.95 


653668 


5 


62 


846887 


45 


i6 


618825 


4 


67 


959825 


-95 


654000 


5 


62 


846000 


44 


\l 


6i4io5 


4 


66 


Q59768 


.95 


654337 


5 


61 


845668 


43 


6i4385 


4 


66 


9597 II 


-95 


654674 


5 


61 


845826 


42 


19 


614665 


4 


66 


959654 


.95 


655oii 


5 


61 


844989 


41 


20 


614944 


4 


65 


959596 


.95 


655348 


5 


61 


844652 


40 


31 


9'6i5228 


4 


65 


9-959589 


.95 


9-655684 


5 


60 


10-844316 


1? 


22 


6i55o2 


4 


65 


959482 


.95 


656o2o 


5 


60 


348980 


23 


615781 


4 


64 


959425 


.95 


656356 


5 


60 


848644 


u 


24 


616060 


4 


64 


959868 


-95 


656692 


5 


59 


843808 


25 


6i6338 


4 


64 


959810 


.96 


657028 


5 


59 


342972 


85 


26 


616616 


4 


63 


959253 


.96 


657864 


5 


.59 


342686 


34 


11 


616894 


4 


68 


959195 


-96 


657699 


5 


.59 


842801 


83 


617172 


4 


62 


959188 


-96 


658o34 


5 


-58 


341966 


32 


19 


617450 


4 


62 


959081 


.96 


658869 


5 


•58 


841681 


3i 


3o 


617727 


4 


62 


959028 


.96 


658704 


5 


•58 


341296 


3o 


3i 


9'6i8oo4 


4 


61 


9-958965 


.96 


9-659089 


5 


•58 


10-340961 


ll 


32 


618281 


4 


61 


958908 


-96 


659878 


5 


57 


840627 


33 


6i8558 


4 


61 


958850 


-96 


659708 


5 


57 


340292 
889958 


U 


34 


618834 


4 


60 


958792 


.96 


660042 


5 


57 


35 


619110 


4 


60 


958784 


-96 


660876 


5 


57 


889624 


25 


36 


619886 


4 


60 


958677 


-96 


660710 


5 


56 


33q2oo 


24 


ll 


619662 


4 


59 


958619 


-96 


661048 


5 


56 


388957 


23 


619988 


4 


59 


958561 


.96 


661877 


5 


56 


888628 


22 


39 


620218 


4 


59 


9585o8 


•97 


661710 


5 


55 


888290 


21 


40 


620488 


4 


58 


958445 


•97 


662043 


5 


55 


337957 


20 


41 


9-620763 


4 


58 


9-958887 


•97 


9-662876 


5 


55 


10-387624 


\t 


42 


621088 


4 


57 


958829 


•97 


662709 


5 


54 


887291 
336958 


43 


621818 


4 


57 


958271 


•97 


668042 


5 


54 


17 


44 


621587 


4 


57 


958218 


•97 


668875 


5 


54 


336625 


I^ 


4i' 


621861 


4 


56 


9581 54 


•97 


668707 


5 


54 


886293 


i5 


4f. 


622135 


4 


56 


958096 
958088 


•97 


664089 


5 


58 


335961 


i4 


% 


622409 


4 


56 


•97 


664371 


5 


58 


335629 


i3 


622682 


4 


55 


957979 


•97 


664708 


5 


53 


335297 


12 


49 


622956 


4 


55 


957921 


•97 


665o85 


5 


53 


884965 


II 


5o 


628229 


4 


55 


957868 


•97 


665366 


5 


52 


884684 


10 


5i 


9-6285o2 


4 


54 


9-957804 


•97 


9-665697 


5 


52 


10 •884803 


« 


52 


628774 


4 


54 


957746 


.98 


666029 


5 


52 


338971 


d 


53 


624047 


4 


54 


957687 
957628 


.98 


666360 


5 


5i 


338640 


I 


54 


624819 


4 


58 


.98 


666691 


5 


5i 


333809 


55 


6245,91 


4 


53 


957570 


.98 


667021 


5 


5i 


332979 


5 


56 


624868 


4 


53 


957511 


.98 


667852 


5 


5i 


882648 


4 


U 


625i35 


4 


52 


957452 


.98 


667682 


5 


5o 


332818 


3 


625406 


4 


52 


957898 
957835 


.98 


668018 


5 


5o 


331987 


2 


59 


625677 
625948 


4 


52 


-98 


668848 


5 5o 


331657 
33i326 


I 


6o 


4»5i 


957276 


-98 


668672 


5 5o 



M^ 




Goeino 


D. 


Sine 


D. 


Cotang. 


D. 


Tang. 



(60 DEGRESS.^ 



SINES AND TANGENTS. ^25 DEGREES.) 



4fi 



M. 


Sine 1 


D. 


Cosine 


D. 1 


Tang. 


D. i 


Cotang. 




o| 


9-625948 


4-5i 


9.957276 


.98 


9-668673 


5'5o ' 


I0'33i327 


60 


I 


626219 


4-5i 


957217 


.98 


669002 


5.49 ! 


330998 


^ 


7 i 


626490 


4-5i 


957158 


.98 


669332 


5-49 


330668 


3 


626760 


4-5o 


957099 


.98 


669661 


5'4q 


33o339 


57 


4 


627030 


4-5o 


957040 


.98 


669991 


5.48 


330009 


56 


5 


627300 


4-5o 


956981 


.98 


670320 


5.48 i 


329680 


55 


6 


627570 


4-49 ■■ 


956921 


•99 


670649 


5.48 ' 


329351 


54 


2 


627840 
628109 
628378 


4-49 ! 


956862 


•99 


670977 
671306 


5'48 


32Q023 


53 


4-49 1 


956803 


•99 


5.47 


328694 


52 


9 


4-48 1 


956744 


•99 


671634 


5.47 


328366 


5i 


10 


628647 


4.48 


956684 


•99 


671963 


5'47 


328037 


5o 


11 


9'6289i6 


4-47 


9.956625 


•99 


9.672291 


5.47 


10-327709 


'^ 


n 


629185 


4-47 i 


956566 


•99 


672619 


5.46 


327381 


a 


629453 


4-47 , 
4-46 . 


9565o6 


•99 


672947 


5.46 


327053 


47 


14 


629721 


956447 


•99 


673274 


5'46 


326726 


46 


i5 


629989 


4-46 


956387 


•99 


673602 


5'46 


326398 


45 


i5 


630257 


4.46 


956327 


•99 


673929 


5'45 


326071 


44 


17 


63o524 


4-46 


g56268 


•99 


674257 


5'45 


325743 


43 


18 


630792 


4-45 


956208 


I'OO 


674584 


5.45 


325416 


43 


»9 


631039 


4-45 


956148 


1. 00 


674910 


5'44 


325090 


41 


30 


63i326 


4-45 


956089 


I -00 


675237 


5-44 


324763 


40 


21 


9-63i593 


4-44 


9.956029 


I'OO 


9-675564 


5-44 


10-324436 


It 


22 


63i85g 


4.44 


955969 


I -00 


675890 


5-44 


324110 


23 


632I2D 


4-44 


955909 


I -00 


676216 


5'43 


323784 


U 


24 


632302 


4-43 


955849 


1. 00 


676543 


5'43 


323457 


25 


632658 


4-43 


955789 


I'OO ! 


676869 


5'43 


323i3i 


35 


26 


632923 


4-43 


955729 


I -00 


677194 


5.43 


322806 


34 


s 


633189 


4-42 


955669 


I'OO 


67-/520 


5.42 


322480 


33 


633454 


4-42 


955609 


I'OO 


67 1846 
670171 


5.42 


322154 


32 


29 


633719 


4-42 


955548 


I'OO 


5.42 


321829 


3i 


3o 


633984 


4-41 


955488 


I'OO 


678496 


5-43 


32i5o4 


3o 


3i 


9-634249 


4-41 


9.955428 


I'OI 


9-678821 


5'4i 


I0'32n79 


29 


32 


634514 


4-40 


955368 


I'OI 


679146 


5'4i 


320854 


28 


33 


634778 


4.40 


955307 


I'OI 


679471 


5'4i 


320529 

320205 


11 


34 


635042 


4.40 


955247 


I'd 


679795 
680120 


5.41 


35 


635306 


4.39 


955186 


I'OI 


5.40 


319880 


25 


36 


635570 


4-39 


955126 


I'OI 


680444 


5'4o 


319556 


24 


37 


635834 


4-3o 
4-38 


955o65 


I'OI 


680768 


5-40 


319232 


23 


38 


636097 


955oo5 


I'OI 


681092 


5'4o 


318908 


22 


39 


636360 


4-38 


954944 


I'OI 


681416 


5'39 


3 1 8584 


21 


40 


636623 


4-38 


954883 


I'OI 


681740 


5.39 


318260 


20 


41 


9-636886 


4-37 


9 954823 


I'OI 


g. 682063 


5.39 


10.317937 


19 


42 


637148 


4-37 


954762 


I'OI 


682387 


5.39 


317613 


19 


43 


6374U 


4-37 


954701 


I-OI 


682710 


5-38 


317290 


17 


44 


637673 


4-37 


95464c 


I'OI 


683o33 


5.38 


316967 


16 


45 


637935 


4-36 


954579 


I'OI 


683356 


5.38 


? 16644 


i5 


46 


638197 


4-36 


954518 


I '02 


683679 


5.38 


3i632i 


14 


47 


638458 


4-36 


954457 


I '02 


684001 


5.37 


3 1 5999 


i3 


48 


638720 


4-35 


954396 
954335 


I '02 


684324 


5'37 


315676 


13 


49 


638981 


4-35 


I '02 


684646 


5.37 


3 1 5354 


II 


5c 


639242 


4-35 


954274 


I '02 


684968 


5.37 


3i5o32 


JO 


5i 


9 -639503 


4-34 


9.954213 


I '02 


9-685290 


5.36 


10-3147/0 

3 I 4388 


I 


5j 


639764 


4-34 


954152 


1.02 


685612 


5.36 


53 


640024 


4-34 


954090 


1.02 


685934 


5.36 


3 1 4066 


7 


54 


640284 


4-33 


954029 


I '02 


686255 


5.36 


3 1 3745 


6 


55 


640544 


4-33 


953968 


1. 02 


686577 
686898 


5-3% 


3 1 3423 


5 


56 


640804 


4-33 


953006 
953845 


1.02 


5.35 


3i3io2 


4 


57 


641064 


4-32 


1-02 


687219 


5.35 


312781 


3 


58 


641324 


4-32 


953783 


1-02 


687540 


5.35 


312460 


3 


59 


64 1 584 


4-32 


953722 


I -03 


687861 


5.34 


3i2i39 
3ii8i8 


I 


60 


64184a 


4-3i 


953660 


I -03 


688182 


5 34 







Cosine 


D. 


Sine 


D. 


Cotang. 


D. 


l&n?. 


_M^ 








(64 


[ DBGR 


ESS.) 









14 



(28 DEGREES.) ^ TABLE OF LOGARITHMIC 



HL 

I 


Sine 


D. 


Cosine I 


). 


Tang. 


D. 


Cotar.g. 





9-641842 


4-3i 


9-953660 I 


o3 


9-688182 


5.34 


io-3ii8i8 1 60 ; 


I 


642 1 01 


4 


3i 


953599 I 


o3 


688502 


5 


34 


811498 


5o 


2 


642860 


4 


3i 


953537 I 


o3 


688828 


5 


34 


811177 


58 


3 


642618 


4 


3o 


953475 I 


o3 


689148 


5 


33 


810857 


57 i 


4 


642877 


4 


3o 


953413 I 


o3 


689468 


5 


83 


3io537 


•X) ' 


5 


643 1 35 


4 


3o 


953352 I 


o3 


689788 


5 


33 


iI02I7 


55 i 


6 


643393 


4 


3o 


953290 I 


o3 


690108 


5 


83 


809897 


54 i 


2 


643650 


4 


29 


953228 I 


08 


690428 


5 


88 


809577 


53 


643908 


4 


29 


953166 I 


o3 


690742 


5 


82 


809258 . 52 


9 


644165 


4 


29 


953104 I 


08 


691062 


5 


32 


.08988 


5i 1 


IC 


644423 


4 


28 


953042 I 


08 


691881 


5 


82 


808619 


5o 


II 


9-644680 


4 


28 


9-952980 1 


04 


9-691700 


5 


3i 


io.3o83oo 


S 


12 


644936 


4 


28 


952018 I 


04 


693019 


5 


81 


807981 


i3 


645 I o3 


4 


27 


952855 I 


04 


692888 


5 


3i 


807662 


47 


14 


645450 


4 


37 


952798 I 


04 


692656 


5 


81 


807844 


46 


i5 


645706 


4 


27 


952781 I 


04 


692975 


5 


3i 


807025 


45 


i6 


645962 


4 


26 


952669 I 


04 


698298 


5 


80 


806707 


44 


\l 


646218 


4 


26 


952606 1 


04 


698612 


5 


80 


806888 


43 


646474 


4 


26 


952544 I 


04 


698980 


5 


3o 


806070 


42 


19 


646729 


4 


25 


952481 I 


04 


694248 


5 


80 


805752 


41 


20 


646984 


4 


25 


952419 


04 


694566 


5 


29 


805484 


40 


21 


9-647240 


4 


25 


9-952856 I 


04 


9-694888 


5 


29 


io-3o5ii7 


3q ' 


i ^? 


647494 


4 


24 


952294 I 


04 


695201 


5 


29 


804799 


38 


! a3 


647749 
648004 


4 


24 


952281 I 


04 


695518 


5 


29 


804482 


37 ' 


i 24 


4 


24 


952168 I 


o5 


695886 


5 


20 


804164 


36 1 


1 25 


648258 


4 


24 


952106 I 


o5 


696153 


5 


28 


808847 


35 


26 


648512 


4 


23 


952048 I 


o5 


696470 


5 


28 


3o858o 


34 . 


?J 


648766 


4 


23 


951980 1 


o5 


696787 


5 


28 


808218 


33 , 


649020 


4 


23 


951917 I 


o5 


697103 


5 


28 


802807 


32 


?9 


649274 


4 


22 


95 1 854 I 


o5 


697420 


5 


27 


3o258o 


3i 


3o 


649527 


4 


22 


951791 I 


o5 


697786 


5 


27 


802264 


3o 


i 3i 


9-649781 


4 


22 


9-951728 I 


o5 


9-698053 


5 


27 


10.801947 


ll 


32 


65oo34 


4 


22 


95 1 665 I 


o5 


698869 


5 


27 


801681 


33 


050287 


4 


21 


951602 I 


o5 


698685 


5 


26 


3oi3i5 


u\ 


34 


65o539 


4 


21 


951389 I 


o5 


699001 


5 


26 


800999 


35 


650792 


4 


21 


951476 I 


o5 


6q98i6 


5 


26 


800684 


25 1 


36 


65 1044 


4 


20 


951412 1 


o5 


699682 


5 


26 


800868 


24 


ll 


65x297 


4 


20 


951849 I 


06 


699947 


5 


26 


8ooo53 


23 


65 1 549 


4 


20 


951286 I 


06 


700268 


5 


25 


299787 


22 


39 


65i8oo 


4 


19 


951222 I 


06 


700578 


5 


25 


299422 


21 


40 


652o52 


4 


19 


951159 I 


06 


700898 


5 


25 


299107 


20 


4i 


9-6523o4 


4 


19 


9951096 I 


06 


9-701208 


5 


24 


10-298792 


^2 

18 


42 


652555 


4 


18 


951082 I 


06 


701528 


5 


24 


298477 


43 


652806 


4 


18 


900968 I 


06 


701887 


5 


24 


298168 


\l 


44 


653o57 


4 


18 


950905 I 


06 


702152 


5 


24 


297848 


45 


6533o8 


4 


18 


950841 I 


06 


702466 


5 


24 


297534 


i5 


1 46 


653558 


4 


n 


950778 I 


06 


702780 


5 


23 


2q7220 


14 


! 47 


6538o8 


4 


17 


950714 I 


06 


708095 


5 


23 


29690J 


!3 


1 48 


654059 


4 


\i 


95o65o I 


06 


708409 


5 


23 


296091 


12 


1 49 


654309 


4 


95o586 I 


06 


•p3723 


5 


23 


296277 


II 


5o 


654558 


4 


16 


95o522 I 


07 


704086 


5 


22 


295964 


10 


5i 


9-654808 


4 


16 


9-950458 I 


07 


9-704350 


5 


32 


15-295650 


g 


52 


655o58 


4 


16 


95o8q4 I 
95o3jo I 


07 


704668 


5 


22 


295337 
295023 


8 


53 


655307 


4 


i5 


07 


704977 


5 


22 


I 


54 


655556 


4 


i5 


950266 I 


07 


705290 


5 


22 


294710 
294807 


55 


6558o5 


f 


i5 


950202 I 


07 


7o56o3 


5 


21 


5 


56 


656o54 


4 


14 


95oi38 I 


07 


705916 


5 


21 


294084 


4 


u 


656302 


4 


14 


950074 I 


07 


706228 


5 


21 


298772 


3 


656551 


4 


14 


gSooio I 


07 


706541 


5 


21 


298459 


2 


59 


656799 


4 


i3 


949045 I 


07 


706854 


5 


21 


298146 


I 


60 


657047 


4i3 


949881 I 


07 


707166 


5-20 


292834 







CJosine 


D. 


Sme I 


). 


Cotang. 


J). 


Tana^. 


M. i 



(63 DEORBE8.) 



SINES AND TANGENTS. (27 DEGREES.) 



46 






Sine 


D. 


Cosine 


D. 


Tfing. 


D. 


Cotang. 


1 


9.657047 
657293 


4-i3 


9-949881 


1-07 


9.707166 


5-20 


10-292884 


60 


I 


4-i3 


949816 


1-07 


707478 


5 


20 


292522 


^ 


3 


657542 


4-12 


949752 


1-07 


707790 


5 


20 


292210 


3 


657700 


4-12 


949688 


I -08 


708102 


5 


•20 


291898 


11 


4 


6500^7 


4-12 


949628 


i-o8 


708414 


5 


19 


291586 


5 


658284 


4-12 


949558 


i-o8 


708726 


5 


19 


291274 


5D 


6 


658531 


4-11 


949494 


1.08 


709087 


5 


19 


290968 


54 


I 


658778 


4-11 


949429 


I -08 


709849 


5 


19 


290651 


53 


659025 


4-11 


949864 


1-08 


709660 


5 


10 


290840 


52 


9 


659271 


4*10 


949800 


1.08 


709971 


5 


18 


290029 


5i 


10 


659517 


4*10 


949235 


I -08 


710282 


5 


18 


289718 


5o 


II 


9 '659763 


4*10 


9-949170 


i-o8 


9.710593 


5 


18 


10-289407 
289096 


49 


12 


660009 


4*09 


949105 


i-o8 


710904 


5 


18 


48 


i3 


660255 


4-09 


949040 


1-08 


711215 


5 


18 


288785 


47 


14 


66o5oi 


4*09 


948975 


1.08 


7ii525 


5 


17 


288475 


46 


i5 


660746 


4-09 


948910 


i-o8 


711886 


5 


17 


288164 


45 


i6 


660991 


4-o8 


948845 


1.08 


712146 


5 


17 


287854 


44 


\l 


661286 


4-o8 


948780 


1-09 


712456 


5 


17 


287544 


43 


661481 


4-o8 


948715 


1-09 


712766 


5 


16 


287284 


42 


19 


661726 


4-07 


948650 


1-09 


718076 


5 


16 


286924 


41 


20 


661970 


4-07 


948584 


1-09 


718886 


5 


i6 


286614 


40 


21 


9-662214 


4*07 


9-948519 


1-09 


9.718696 


5 


16 


10.286804 


89 


22 


662459 


4-07 


948454 


1-09 


714005 


5 


16 


283995 


88 


23 


662703 


4-o6 


948888 


1-09 


714814 


5 


i5 


285686 


37 


24 


662946 


4-o6 


948828 


1-09 


714624 


5 


i5 


285876 


36 


25 


663190 


4-o6 


948257 


1-09 


714933 


5 


i5 


285067 


35 


26 


663483 


4-o5 


948192 


1-09 


715242 


5 


i5 


284758 


34 


11 


663677 


4-o5 


948126 


1-09 


7i555i 


5 


14 


284449 


33 


663920 


4-o5 


948060 


1-09 


7i586o 


5 


14 


284140 


32 


?9 


664163 


4-o5 


947995 


1-10 


716168 


5 


14 


288882 


3i 


So 


664406 


4-04 


947929 


I-IO 


716477 


5. 


14 


283523 


3o 


3i 


9.664648 


4-04 


9-947863 


I'lO 


9.716785 


5. 


14 


10.288215 


29 


32 


664891 


4-04 


947797 
9477^1 


1-10 


717098 


5. 


i3 


282907 


28 


33 


665 I 33 


4-o3 


I-IO 


717401 


5. 


i3 


282399 


27 


34 


665375 


4-o3 


047665 


I-IO 


717709 


5. 


i3 


282291 


26 


35 


665617 


4-o3 


947600 


I-IO 


718017 


5. 


i3 


281988 


25 


36 


665859 


4-02 


947538 


1-10 


718825 


5. 


i3 


281670 


24 


ll 


666100 


4-02 


947467 


1-10 


718688 


5. 


12 


281867 


23 


666342 


4-02 


947401 


l-IO 


718940 


5. 


12 


281060 


22 


39 


666583 


4-02 


947835 


1-10 


719248 


5. 


12 


280752 


21 


40 


666824 


4-01 


947269 


1-10 


719555 


5. 


12 


280445 


20 


41 


9-667065 


4'oi 


9-947208 


I-IO 


9.719862 


5. 


12 


10-280188 


\t 


42 


667805 


4*01 


947186 


1-11 


720169 


5. 


11 


279881 


43 


667546 


4*01 


947070 


l-ll 


720476 
720783 


5. 


11 


279524 


'7 


44 


667786 


4*00 


947004 


1-11 


5- 


II 


270217 


16 


45 


668027 


4*00 


946937 


l-II 


721089 


; 


11 


278911 ! 1 5 


46 


668267 


4-00 


946871 


1-11 


721896 


5. 


II 


278604 14 


% 


6685o6 


^■99 


946804 


1-11 


721702 


5. 


10 


278298 


i3 


668746 


^•99 


946788 


I'll 


722009 


5- 


10 


277685 


la 


49 


668986 


^*99 


946671 


l-Il 


722813 


5. 


10 


II 


5o 


669225 


3-99 


946604 


I-ll 


722621 


5. 


10 


277879 


10 


5i 


9-669464 


3.98 


9.946538 


1-11 


9.722927 


5. 


10 


10.277073 


? 


52 


669708 


3-98 


946471 


I-II 


728282 


5. 


09 


276768 


53 


669942 


3-98 


946404 


l-ll 


723538 


5. 


09 


276462 7 i 


54 


670181 


3-97 


946887 


l-I] 


728844 


5. 


09 


276156 


6 


55 


670419 


^97 


946270 


1-1] 


724149 


5. 


09 


275851 


5 


56 


670658 




?'97 


946208 


1-12 


724454 


5. 


S 


275546 


4 


u 


670896 
671184 


X*97 


946186 


1-12 


724759 
725o65 


5 


275241 


3 


3-96 


946069 


1-12 


5 


08 


274935 
274631 


3 


59 


671872 


3-96 


946002 


1-12 


725369 


5 


08 


I 


6o 

U-.J 


671609 


3-96 


945935 


1-12 


725674 


5.08 


274826 





Cofline 


D. 


Sine 


D. 


Cotang. 


D. 


Tang. 


M. 



(62 DEGRSJIS.) 



i6 



(28 DEGREES.) A TABLE OF LOGARITHMIC 



M. 


Biiio 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 





9.671609 


3.96 


9 •945085 

943868 


1*13 


9.725674 


5 08 


10.274336 


1 


, 671847 


3 


.95 


I-I3 


725979 


5 08 


374031 


So 


9 


672084 


3 


.95 


943800 


1-13 


! 726284 


5 


• 07 


3787x6 


58 


3 


672321 


3 


.95 


945788 


I.I3 


,26588 


' 5 


.07 


2784x2 57 
273108 56 


4 


672558 


3 


.93 


945666 


I'13 


736893 


5 


•07 


5 


672795 
673082 


3 


.94 


945598 
945531 


I.X2 


737197 


5 


•07 


272803 55 


6 


3 


.94 


1.13 


737501 


5 


•07 


272499 54 
272195 53 


I 


673268 


8 


.94 


943464 


113 


737805 


5 


.06 


673505 


3 


.94 


945896 


1-13 


728109 


5 


•06 


27X89X 53 


9 


673741 


3 


.93 


945328 


I.l3 


738413 


5 


.06 


271588 


5i 


10 


6^39- 


3 


•93 


945261 


113 


7387x6 


5 

1 


.06 


271284 


5. 


II 


9 674313 


8 


•93 


9.945198 


113 


9.729020 


5 


.06 


10.270980 


S 


13 


6744/18 


8 


92 


945125 


i-i3 


729823 


1 ^ 


.o5 


270677 


i3 


674684 


3 


93 


945o58 


ii3 


729626 


' 5 


•o5 


270874 


47 


14 


674919 


3 


92 


944990 


i.i3 


729929 
780283 


i 5 


.05 


270071 


46 


i5 


675x55 


3 


92 


944922 


i.i3 


1 5 


.05 


269767 


45 


i6 


675390 


3 


91 


944854 


i.i3 


780535 


5 


.o5 


269465 


44 


\l 


675634 


3 


91 


944786 


i.i3 


780888 


5 


•04 


269162 
268859 


43 


575359 


3 


91 


944718 


i.x3 


781x41 


5 


04 


43 


^9 


676094 


3 


91 


944650 


i.i3 


78x444 


5 


.04 


268556 


41 


20 


676328 


3 


90 


944582 


1.X4 


78x746 


5 


•04 


268254 


40 


31 


9.676563 


3 


90 


9.944514 


1-14 


9-782048 


5 


04 


10.267952 


^ 


33 


676796 


3 


90 


944446 


114 


73235X 


5 


o3 


267649 


33 


677080 


3 


90 


944877 


1.14 


732653 


5 


o3 


267847 


u 


24 


677264 


8 


°9 


944809 


1-14 


782935 


5 


o3 


267045 


25 


677498 
6777^1 


8 


89 


944341 


1.14 


788257 


5 


o3 


266743 


35 


36 


3 


89 


944172 


1.14 


788558 


5 


o3 


266442 


34 


11 


677964 


3 


88 


944104 


1. 14 


788860 


5 


03 


266140 


33 


678197 


3 


88 


9/r'io86 


X.14 


784162 


5 


03 


265888 


32 


29 


678430 


3 


88 


943067 
943899 


1.14 


784468 


5 


02 


265537 


3i 


3o 


678663 


3 


88 


1. 14 


734764 


5 


02 


265236 


3o 


3i 


9.678895 


3 


87 


9.948880 


X.14 


9 -735066 


5 


03 


10.264984 


^ 


33 


679128 


8 


87 


948761 


1.X4 


735867 


5 


02 


264688 


33 


679860 


8 


87 


943698 


x.i5 


785668 


5 


OX 


264882 


U 


34 


679393 


3 


87 


943624 


i.i5 


735969 


5 


01 


264081 


35 


679824 


3 


86 


948555 


x.i5 


786269 


5 


01 


26878 X 


25 


36 


68oo56 


3 


86 


943486 


i.i5 


786570 


5 


01 


268480 


24 


37 


680288 


3 


86 


948417 


i.i5 


736871 


5 


OX 


268129 


33 


38 


680319 


3. 


85 


948848 


x.i5 


787171 


5 


00 


262829 


23 


39 


680700 


3. 


85 


948279 


i.i5 


787471 


5 


00 


262529 


21 


40 


680982 


3 


85 


943210 


x.x5 


737771 


5 


00 


262239 


30 


41 


9.681213 


8. 


85 


9.948141 


x.i5 


9 -78807 1 


5 


00 


10.261939 


:? 


42 


681443 


8. 


84 


948072 


i.i5 


788871 


5 


00 


261629 


43 


681674 


3- 


84 


948008 


1x5 


788671 


4- 


99 


26x829 17 j 


44 


681905 


3. 


84 


942984 


x.i5 


738971 


4- 


99 


26x029 j 16 i 


45 


682135 


3. 


84 


942864 


x.i5 


789271 


4- 


99 


260729 ; i5 i 


46 


682365 


3- 


88 


942795 


X.16 


789570 


4 


99 


260480 ! 14 ! 


il 


682595 


3. 


88 


942726 


i.x6 


789870 


4 


99 


260180 j 3 i 


682825 


8. 


88 


942656 


I -16 


740160 


4- 


99 


259881 \ la 1 


49 


683o55 


3. 


88 


942587 


1-16 


74046S 


4 


9^ , 


259532 


II 


5o 


688284 


3- 


82 


9435x7 


1.16 


740767 


4 


98 


259288 


le 


5i 


9 6835i4 


3. 


82 


9.942448 


1-16 


9.741066 


4 


98 


10.258934 


I 


52 


683743 


8. 


82 


942878 


I -16 


741865 


4- 


98 


258635 


53 


683972 


8. 


82 


942808 


1. 16 


741664 


4- 


98 


258386 


I 


54 


684201 


8. 


81 


942289 


i-i6 


741962 


4- 


97 


258o38 


55 


684480 


3. 


81 i 


942169 


1.16 


742261 


4- 


97 


357789 


5 


56 


684658 


3. 


81 i 


942099 


1. 16 


742559 


4- 


97 


257441 


4 


5^ 


684887 


3- 


80 : 


942029 


i-i6 ' 


743858 


4- 


97 


257x43 ■ 3 J 


685ii5 


3. 


80 : 


941959 


x.i6 ' 


7481 56 


4- 


97 


356844 


2 


59 


685343 


3. 


So i 


941889 


I.X7 


743454 


4- 


97 


256546 


I 


60 


685571 


3.80 


941819 


1. 17 


748753 


4-96 


356348 1 







Coeiue 


D. ! 


Sine 


D. 


Cotang. 


D. 1 


Tang. 1 


M. 



(61 DEaRESS.) 



I 



SINES AND TANGENTS. (29 DEGREES.; 



47 



M. 


Sine 1 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


r 
1 


o 


9-685571 


3.80 


9-941819 


I-I7 


9-743752 


4.96 


10-256248 60 


I 


685799 


3- 


79 


941749 


1. 17 


744o5o 


4 


.96 


255950 5o 
255652 i 58 


2 


686027 


3. 


79 


941679 


117 


744348 


4 


.96 


3 


686254 


3- 


79 


941609 


1. 17 


744645 


4 


.96 


255355 ' 57 
255o57 ; 56 


4 


686482 


3- 


]l 


941539 


1.17 


744943 


4 


96 


5 


686709 


3- 


941469 


1. 17 


745240 


4 


9^ 


254760 i 55 


6 


686936 


3- 


78 


941398 


117 


745538 


4 


9^ 


254462 54 


I 


687163 


3- 


78 


941328 


I-I7 


745835 


4 


9? 


254165 53 


687^89 
687616 


3. 


78 


941258 


1.17 


746132 


4 


9^ 


253868 


52 


9 


3- 


77 


941187 


1. 17 


746429 


4 


9^ 


253571 


5i 


10 


687843 


3. 


77 


941 117 


1.17 


746726 


4 


95 


253274 


5o 


II 


9-688069 
688295 


3 


77 


9-941046 


1.18 


9-747023 


4 


94 


10-252977 


% 


12 


3. 


77 


940975 


1.18 


747319 


4 


94 


252681 


i3 


688521 


3 


76 


940905 


1. 18 


747616 


4 


94 


252384 


47 


14 


688747 


3 


76 


940834 


1.18 


747913 


4 


94 


252087 


46 


i5 


688972 


3 


76 


940763 


i-i8 


748209 


4 


94 


251791 


45 


\6 


689198 


3 


76 


940693 


1. 18 


7485o5 


4 


93 


251495 


44 


\l 


689423 


3 


75 


940622 


1. 18 


748801 


4 


93 


251199 
250900 


43 


689648 


3 


75 


94055 I 


1x8 


749097 


4 


93 


42 


^ 


689873 


3 


7? 


940480 


i-i8 


749393 


4 


93 


2 50607 


41 


20 


690098 


3 


75 


940409 


1. 18 


749689 


4 


93 


25o3ii 


40 


21 


9690323 


3 


74 


9 -940338 


1. 18 


9-749985 


4 


93 


io.25ooi5 


^? 


22 


690548 


3 


74 


940267 


1-18 


750281 


4 


92 


249719 


23 


690772 


3 


74 


940196 


i-i8 


750576 


4 


92 


249424 


3l 


24 


690996 


3 


74 


940125 


1-19 


750872 


4 


92 


249128 


36 


25 


691220 


3 


73 


940054 


1.19 


751167 


4 


02 


248833 


35 


26 


691444 


3. 


73 


939982 


119 


751462 


4 


92 


248538 


34 


27 
28 


691668 


3 


73 


939911 


119 


751757 


4 


92 


248243 


33 


691892 


3 


73 


939840 


I 19 


752o52 


4 


91 


247948 


32 


?9 


6921 1 5 


3 


72 


939768 


1. 19 


752347 


4 


9» 


247653 


3i 


3o 


692339 


3 


72 


939697 


119 


752642 


4 


91 


247358 


3o 


3i 


9 692562 


3 


72 


9-939625 


1-19 


9.752937 


4 


9» 


10.247063 


?? 


32 


692785 


3 


71 


939554 


I -19 


75323i 


4 


91 


246769 


33 


693008 


3 


71 


939482 


1. 19 


753526 


4 


91 


246474 


\L 


34 


693231 


3 


7» 


939410 


I 19 


753820 


4 


90 


246180 


35 


693453 


3 


71 


939339 


1-19 


7541 1 5 


4 


90 


245885 


25 


36 


693676 


3 


70 


939267 


1-20 


754400 


4 


90 


245591 


24 


U 


693898 


3 


70 


939195 


1-20 


754703 


4 


90 


245297 


23 


694120 


3 


70 


939123 


1-20 


754997 


4 


90 


245oo3 


22 


39 


694342 


3 


70 


939052 


1-20 


755291 


4 


90 


244709 
244415 


21 


40 


694564 


3 


69 


938980 


1-20 


755585 


4 


89 


20 


41 


9 -694786 


3 


69 


9.938908 


1-20 


9.755878 


4 


^9 


10-244122 


\% 


42 


695007 


3 


69 


938836 


1-20 


756172 


4 


^9 


243828 


43 


695229 


3 


69 


938763 


1-20 


756465 


4 


?9 


243535 


\l 


44 


695450 


3 


68 


938691 


1-20 


756759 


4 


^9 


243241 


45 


695671 


3 


68 


938619 


1-20 


757052 


4 


^ 


242948 


i5 


46 


695892 


3 


68 


938547 


1-20 


757345 


4 


88 


242655 


14 


f 47 
48 


6961 i3 


3 


68 


938475 


1-20 


757638 


4 


88 


242362 


i3 


696334 


3 


67 


938402 


1-21 


757931 


4 


88 


242069 


13 


^9 


690554 


3 


67 


938330 


1-21 


758224 


4 


88 


241776 


11 


5o 


690775 


3 


67 


938258 


1-21 


758517 


4.88 


241483 


ID 


5i 


9-696995 


3 


t>7 


9-938185 


i'21 


9.758810 


4.88 


10-241190 


I 


53 


697215 


3 


-66 


9381 i3 


1-21 


759102 


4.87 


240898 


53 


697435 


3 


-66 


938040 


1-21 


759395 


4-87 


240605 


I 


54 


607654 


3 


-66 


937967 


I-2I 


759687 


4.87 


24o3i3 


55 


697874 
698094 


3 


-66 


937895 


1-21 


759979 


4.87 


240021 


i> 


56 


3 


.65 


937822 


I-2I 


760272 


4.87 


239728 


4 


U 


698313 


3 


-65 


937749 


1-21 


760564 


4.87 


239436 


3 


698532 


3 


-65 


937676 


I-2I 


760856 


4.86 


239144 

238852 


a 


59 


698751 


3 


-65 


937604 


' -21 


761148 


4.86 


I 


60 


698970 


3 

_ J 


.64 


937531 


I-2I 


761439 


4-86 


238561 





CoBine 


0. 


' Sine 


D. 


Cotang. 


1 D. 


Tang. 


_M._^ 



27 



(60 DBORBBS.) 



48 


(SO DEGREES.) A TABLE 


OF L0GARlTHMi:5 




M. 


Bino 


D. 


Cosine 1 D. 


Tan^. 


D. 


Ootan^f. 


~ 


o 


9 •698970 
699189 


3.64 


9-937531 1 I 


•21 


9-761439 


4-86 


10-238561 


60 


I 


3 


.64 


937458 I 


•22 


761781 


4 


-86 


288269 


5o 


3 


699407 


3 


•64 


937385 I 


•22 


762028 


4 


-86 


287977 56 
2876?^ 57 


3 


699626 


3 


64 


937312 I 


•22 


762814 


4 


-86 


4 


699844 


3 


63 


987288 I 


-22 


762606 


4 


•85 


387894 


56 


5 


700062 


3 


63 


937165 I 


■22 


762897 


4 


-85 


287103 


55 


6 


700280 


3 


63 


987092 I 


-22 


768188 


4 


•85 


286812 


54 


7 


7P0498 


3 


63 


987019 I 


•22 


7634-9 


4 


-85 


236521 


53 


700716 


3 


63 


986046 I 


-22 


768770 


4 


-85 


236280 


52 


9 


700933 


3 


62 


986872 I 


-22 


764061 


4 


•85 


285939 
235648 


5i 


10 


7oii5i 


3 


62 


986799 I 


-22 


764352 


4 


-84 


5o 


II 


9-701368 


3 


62 


9.986725 1 


22 


9-764643 


4 


-84 


10.235357 


it 


12 


701585 


3 


62 


986652 I 


-23 


764988 


4 


84 


285067 


i3 


701802 


3 


61 


986578 ' I 


-23 


765224 


4 


84 


284776 
284486 


s 


14 


702019 


3 


61 


9865o5 I 


23 


765514 


4 


84 


'i 


702236 


3 


61 


98648 1 I 


23 


7658o5 


4 


84 


284195 


45 


i6 


702452 


3 


61 


986857 1 


23 


766095 


4 


84 


288905 


44 


\l 


702669 
702885 


3 


60 


986284 I 


-23 


766385 


4 


83 


2836i5 


43 


3 


60 


986210 I 


23 


766675 


4 


83 


283825 


42 


19 


7o3ioi 


3 


60 


986186 1 


23 


766965 


4 


88 


233o35 


41 


20 


703317 


3 


60 


986062 I 


23 


767255 


4 


83 


282745 


40 


21 


9 -703 533 


3 


59 


9.985988 I 


23 


9-767545 


4 


83 


10-282455 


39 
38 


22 


703749 


3 


59 


985914 I 


23 


767884 


4 


88 


282166 


23 


703964 


3 


59 


985840 I 


23 


768124 


4 


82 


281876 


11 


24 


70417? 
704395 


3 


59 


935766 I 


24 


768418 


4 


82 


23 1 587 


25 


3 


ll 


985692 I 


24 


768708 


4 


82 


281297 
281008 


35 


26 


704610 


3 


935618 I 


24 


768992 


4 


82 


34 


U 


704825 


3 


58 


935543 I 


24 


769281 


4 


82 


280719 


33 


7o5o4o 


3 


58 


985469 1 
935893 I 


24 


769570 


4 


82 


280480 


32 


29 


7o5254 


3 


58 


24 


769860 


4 


81 


280140 


3i 


3o 


705469 


3 


57 


935320 I 


24 


770148 


4 


81 


22q852 


3o 


3i 


9 •705683 


3 


^7 


9.985246 I 


24 


9-770487 


4 


81 


10-229568 


2Q 

28 


32 


705898 


3 


?7 


985171 I 


24 


770726 


4 


81 


229274 


33 


7061 1 2 


3 


57 


985097 I 


24 


771015 


4 


81 


228985 


27 


34 


706326 


3 


56 


935022 I 


24 


771808 


4 


81 


228697 
228408 


26 


35 


706539 
706753 


3 


56 


984948 I 


24 


771592 


4 


81 


25 


36 


3. 


56 


984878 I 


24 


771880 


4 


80 


228120 


24 


u 


706967 


3 


56 


984798 I 


25 


772168 


4 


80 


227882 


23 


707180 


3. 


55 


984728 I 


25 


772457 


4- 


80 


227543 


22 


39 


707393 


3 


55 


984649 J 


25 


772745 


4 


80 


227255 


21 


40 


707606 


3. 


55 


984574 I 


25 


778083 


4 


80 


226967 


20 


41 


9.707819 


3. 


55 


9-984499 I 


25 


9-778821 


4- 


80 


10.226679 


[I 


42 


708032 


3 


54 


984424 I 


25 


778608 


4- 


79 


226892 


43 


708245 


3- 


54 


984849 I 


25 


778896 


4- 


79 


226104 1 17 
2258i6 ! 16 


44 


708458 


3 


54 


984274 » 


25 


774184 


4- 


79 


^f 


708670 


3 


54 


984195 I 


25 


774.;7i 


4 


79 


225529 i5 


46 


708882 


3 


53 


984123 i I- 


25 


774759 


4 


79 


225241 I4 


% 


709094 


3- 


53 


984048 I 


25 


775046 


4- 


79 


224954 I J 


709306 


3 


53 


988975 i I 


25 


775838 


4- 




224667 '2 1 


i^ 


709518 


3. 


53 


933898 1 I 


26 


775621 


4- 


78 


224879 


-I ! 


5o 


709730 


3. 


53 


988822 I 


26 


775908 


4- 


78 


224092 


10 


5i 


9.709941 


3 


52 


9-933747 I- 


26 


9-776195 


4- 


78 


io.2238o5 


i 


52 


;ioi53 


3. 


52 


988671 I- 


26 


776482 


d- 


78 


2235i8 


53 


710364 


3 


52 


983596 I 


26 


776769 


4- 


78 


228281 


l\ 


54 


710575 
71.0786 


3. 


52 


938520 I 


26 


777055 


4- 


78 


222945 


55 


3 


5i 


988445 ! I- 


26 


777342 


4- 


78 


222658 


5 1 


56 


7x0997 
711 208 


3- 


5i 


988860 1 - 
933298 I • 


26 


777628 


4^ 


77 


222872 


4 


U 


3 


5i 


26 


777915 


A- 


77 


222085 


3 


711419 


3. 


5i 


988217 I- 


26 


778201 


4- 


77 


221799 

22l5l2 


3 


59 


711629 


3 


5o 


988141 I* 


26 


778487 


4- 


77 


I 


60 


711839 


3.50 


988066 I « 


26 


778774 


4-77 


221226 







Cosine 


D. 


Sine r 


►, 


Cotang. 


Tang. M. 










(59 DE 


GRI 


BE8.) 













SINES AND TANGENTS. 


C31 DEGREES.) 


49 


M. 




Sine 


D. 


Cosine 


D. • 


Tang. 


D. 


CotftTlff. 




9-711839 


3-5o 


9-933066 


1-26 


9-778774 


4-77 


10-221226 


60 


I 


7i2o5o 


3-50 


982990 


1-27 


779060 


4-77 
4-76 


220940 5o 
220654 58 


2 


712260 


3-50 


932914 


1-27 


779346 


3 


712469 


3.49 


982838 


1.27 


779682 


4-76 


220868 57 
220082 5o 


4 


712679 
7128B9 
713098 


3-49 


982762 


1.27 


779918 


4-76 


5 


3-49 


982685 


1-27 


780208 


4-76 


2x9797 ; 55 


6 


3-49 


982609 


1-27 


780489 


4-76 


2195XX 


54 


i 


7i33o8 


3.49 


93253w 


1-27 


780775 


4-76 


2X0225 
2x8940 


53 


7i35i7 


3.48 


982457 


1.27 


781060 


4-76 


53 


9 


713726 


3-48 


932880 


1-27 


781846 


4-75 


2x8654 


5i 


10 


713935 


3.48 


982804 


1-27 


781681 


4-75 


2x8369 


5o 


II 


9.714144 


3.48 


9-982228 


1.27 


9-781916 


4-7^ 


10-218084 


49 
48 


la 


714352 


3-47 


982151 


1-27 
1.28 


782201 


4-75 


217799 


i3 


714561 


3-47 


982075 


782486 


4-75 


2x7514 


% 


14 


71476Q 
714978 


3-47 


981998 


1-28 


782771 


4-75 


2x7229 


i5 


3-47 


981921 


1-28 


788056 


4-75 


216944 


45 


i6 


7i5i86 


3-47 


98x845 


1-28 


788841 


4-75 


216659 


44 


\l 


715394 


3.46 


981768 


1-28 


788626 


4-74 


216874 


43 


7i56o2 


3.46 


981691 


1-28 


788910 


4-74 


2x6090 


42 


»9 


715809 


3.46 


981614 


1-28 


784195 


4-74 


2i58o5 


41 


20 


716017 


3.46 


981587 


1-28 


784479 


4-74 


2X5521 


40 


ai 


9-716224 


3.45 


9-981460 


1.28 


9-784764 


4-74 


io-2i5286 


39 


22 


716432 


3.45 


981888 


1.28 


785048 


4-74 


214952 


88 


23 


716639 


3.45 


981806 


1.28 


785832 


4-73 


2x4668 


ll 


24 


716846 


3-45 


981229 


1-29 


7856x6 


4-73 


2x4884 


25 


717053 


3.45 


981152 


1-29 


785900 


4-73 


214x00 


35 


26 


717259 


3.44 


98x075 


1-29 


786x84 


4-73 


2x38x6 


34 


s 


717466 


3-44 


980998 


1-29 


786468 


4-73 


2x3532 


33 


717673 


3-44 


980921 


1-29 


786752 


4-73 


218248 


32 


?9 


717870 
7180S5 


3-44 


980848 


1-29 


787086 


4-73 


212964 


3i 


3o 


3.43 


980766 


1-29 


787819 


4-72 


212681 


3o 


3i 


9-718291 


3.43 


9-980688 


1-29 


9-787608 


4-72 


10-2x2897 


It 


32 


718497 


3-43 


980611 


1-29 


787886 


4.72 


2X2XX4 


33 


718703 


3-43 


98o538 


1-29 


788170 


4-72 


2I1880 


27 


34 


718909 


3.43 


980456 


1-29 


788453 


4-72 


2xx547 


26 


35 


719114 


3.42 


930878 


1-29 


788786 


4-72 


2x1264 


25 


36 


719320 


3.42 


980800 


1-30 


7890x9 


4-72 


2x0981 


24 


ll 


719525 


3-42 


980228 


i-3o 


789802 


4-71 


2x0698 


23 


719730 


3-42 


980145 


i-3o 


789585 


4-71 


2X04x5 


22 


39 


719935 


3-41 


980067 


i-3o 


789868 


4-71 


2x0x82 


21 


40 


720140 


3.41 


929989 


I -30 


7901 5i 


4-71 


209849 


20 


41 


9-720345 


3-41 


9'9299Xi 


1.30 


9-790488 


4-71 


10-209567 


19 


42 


720549 


3-41 


929888 


1-30 


7907x6 


4-71 


209284 


43 


720754 


3-40 


929755 


1.80 


790999 


4-71 


209001 


\l 


44 


720958 


3-40 


929677 


1-30 


79x281 


4-71 


2007x9 


45 


721162 


3.40 


929599 


1-80 


79x563 


4-7® 


208487 


i5 


46 


72i366 


3-40 


929521 


1-80 


791846 


4-70 


2081 54 


14 


47 


721570 


3-40 


929442 


X.30 


792x28 


4-70 


207872 


i3 


4S 


721774 


3.39 


929864 


1-81 


7924x0 


4-70 


207590 


12 


49 


721978 
722181 


3-39 


929286 


1-81 


792692 


4-70 


207808 


11 


5o 


3-39 


929207 


1-31 


792974 


4-70 


207026 


10 


5i 


9-722385 


3-39 


9-929129 


1-31 


9-798256 


4-70 


10-206744 


t 


52 


722588 


3.39 


929060 


1-31 


798538 


4.69 


206462 


53 


722791 


3-38 


928972 


i-3i 


798819 


4.69 


206181 


I 


54 


722994 


3-38 


928898 


i-3i 


794x01 


4.69 


205899 


55 


723197 


3-38 


9288x5 


i.3i 


794888 


4-69 


2o56i7 


5 


56 


723400 


3-38 


928786 


i-8i 


794664 


4-69 


205336 


4 


^ 


7236o3 


3-37 


928657 


i-3i 


794945 


4-69 


2o5o55 


3 


7238o5 


3-37 


928578 


i-3i 


796227 


4.65 


204773 


a 


59 


724007 


3-37 


928499 


1-31 


795508 


4.68 


204492 


I 


6o 


724210 


3-37 


928420 


1-81 


795789 


4-68 


2042 II 







CoBine 


D. 


Sine 


D. 


Cotang. 


1 D. 


Tang. 


_r , 






(58 


t DEGR 


SIS.) 









50 



(32 DEGREES.) A TABLE OF LOGARITHMIC 



M. 


Sine 


D. 


Cosine D. 


Tang. 


D. 


Cotang. 


1 
1 





9-724210 


3.37 


9-928420 I 


32 


9-795789 


4-68 


10-204211 


60 1 


I 


724412 


3 


37 


928842 I 


32 


796070 


4 


68 


208980 


u 


2 


724614 


3 


36 


928263 I 


82 


796851 


4 


68 


208649 


3 


724816 


3 


86 


928188 I 


82 


796682 


4 


68 


208868 


57 


4 


725017 


3 


86 


928104 I 


32 


796913 


4 


68 


208087 


56 


5 


725219 


3 


36 


928025 I 


32 


797194 


4 


-68 


202806 


55 


6 


725420 


8 


85 


927946 I 


32 


797475 


4 


-68 


202525 


54 


I 


725622 


3 


35 


927867 I 


82 


797755 


4 


68 


202245 


53 


725823 


3 


35 


927787 I 
927708 1 


32 


798086 


4 


67 


201964 


52 


9 


726024 


3 


35 


32 


798816 


4 


67 


201684 


5i 


n 


726225 


3 


3a 


927629 I 


82 


798596 


4 


67 


201404 


5o 


II 


9-726426 


3 


34 


9-927549 I 


82 


9-798877 


4 


67 


IO-20II23 


49 


12 


726626 


3 


34 


927470 I 


33 


799157 


4 


67 


200848 


48 


i3 


726827 


3 


34 


927890 I 


83 


799437 


4 


67 


2oo568 


47 


14 


727027 


3 


34 


927810 I 


83 


799717 


4 


67 


200288 


46 


i5 


727228 


3 


84 


927281 I 


88 


799997 
800277 


4 


66 


200008 


45 


i6 


727428 


3 


33 


927151 I 


83 


4 


66 


199728 


44 


\l 


727628 


3 


33 


927071 1 


88 


800557 


4 


66 


199443 


43 


727828 


8 


33 


926991 I 


83 


800886 


4 


66 


I99r64 


42 


19 


728027 


3 


33 


92691 1 I 


83 


801 1 16 


4 


66 


198884 


41 


20 


728227 


3 


33 


926881 I 


38 


801896 


4 


66 


198604 


40 


21 


9-728427 


3 


32 


9-926751 I 


88 


9-801675 


4 


66 


10-198825 


39 


22 


728626 


3 


32 


926071 I 


38 


801955 


4 


66 


198045 


88 


23 


728825 


3 


32 


926591 I 


38 


802284 


4 


65 


197766 


u 


24 


729024 


3 


32 


9265ii I 


34 


8o25i3 


4 


65 


197487 
197208 


25 


729323 


3 


3i 


926481 I 


34 


802792 


4 


65 


35 


26 


729422 


3 


3i 


92685i I 


34 


808072 


4 


65 


196928 


34 


11 


729621 


3 


3i 


926270 I 


34 


8o885i 


4 


65 


196649 


33 


729820 


3 


3i 


926190 I 


34 


808680 


4 


65 


196870 


32 


29 


780018 


3 


3o 


926110 I 


34 


808908 


4 


65 


196092 


3i 


3o 


730216 


3 


3o 


926029 1 


34 


804187 


4 


65 


195813 


3o 


3i 


9 -73041 5 


3 


3o 


9-925949 I 


3/ 
3^ 
"^4 


9-804466 


4 


64 


10-195584 


ll 


32 


73o6i3 


3 


3o 


925868 I 


804745 


4 


64 


195255 


33 


780811 


3 


3o 


925788 1 


34 


8o5o23 


4 


64 


I94Q77 
194698 


27 


34 


781009 


3 


29 


925707 I 


34 


8o53o2 


4 


64 


26 


35 


781206 


3 


29 


925626 I 


34 


8o558o 


4 


64 


194420 


25 


36 


781404 


3 


29 


925545 I 


35 


8o5859 


4 


64 


194141 


24 


ll 


781602 


3 


29 


925465 I 


35 


806187 
80641 5 


4 


64 


198868 


23 


781799 


8 


29 


925384 I 


35 


4 


63 


198585 


22 


39 


781996 


3 


28 


925808 I 


35 


806698 


4 


63 


198807 


21 


40 


782198 


3 


28 


923222 I 


35 


806971 


4 


63 


198029 


20 


41 


9-782890 


3 


28 


9-925i4i i 


35 


9-807249 


4 


68 


10-192751 


10 


42 


782587 


3 


28 


925o6o I 


35 


807527 


4 


63 


192478 


43 


782784 


3 


28 


924079 I 


35 


807805 


4 


68 


192195 


17 


44 


782980 


3 


27 


924897 I 


35 


808083 


4 


63 


191917 


16 


45 


733177 


3 


27 


924816 I 


35 


808861 


4 


63 


191689 


i5 


46 


788878 


3 


27 


924735 I 


86 


808688 


4 


62 


191862 


14 


47 


788560 
788765 


3 


27 


924654 I 


36 


808916 


4 


62 


191084 


i3 


48 


3 


27 


924572 1 


36 


809198 


4 


62 


190807 


11 


49 


788961 


3 


26 


924491 I 


36 


809471 


4 


62 


190529 


11 


5o 


784157 


3 


26 


924409 I 


36 


809748 


4 


62 


190252 


10 


5i 


0- 784853 


3 


26 


9-924828 I 


36 


9-8ioo25 


4 


62 


10-189975 


? 


52 


734549 


3 


26 


924246 I 


36 


810802 


4 


62 


189698 


53 


734744 


3 


25 


924164 I 


36 


8io58o 


4 


62 


189420 


7 


54 


734939 


3 


25 


924088 I 


36 


810857 


4 


62 


189143 


6 


55 


785i35 


3 


25 


924001 I 


36 


811184 


4 


61 


188866 


5 


56 


785830 


3 


25 


928919 I 


36 


811410 


4 


61 


188590 


4 


U 


785525 


3 


25 


928887 I 


86 


81 1687 


4 


61 


188818 


3 


735719 


3 


24 


928755 . I 


37 


81 1964 


4 


61 


188086 


2 


59 


735914 


3.24 


928678 I 


37 


812241 


4 


61 


187750 
187483 


I 


60 

1 


786109 


3-24 


923591 I 


37 


812517 


4-6i 



M. 


Cofliue 


D. 


Sine I 


). 


Cotang. 


D. 


Tanff. 



(57 DEGREES.) 



snrsa and tangents. (33 degrees.) 



51 



^M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotan^- 


60 





9-736109 
7363o3 


3.24 


9.923591 


1.37 


9.812517 


4-6i 


10.187482 


I 


3-24 


923509 


1.37 


812794 


4-6i 


187206 


U 


a 


736498 


3-24 


923427 
923345 


i.3i 


818070 


4.61 


1 86980 


3 


736692 
736886 


3-23 


1.87 


818847 


4-60 


186653 


U 


4 


3-23 


923263 


1.87 


818628 


4.6c 


186877 


5 


737080 


3.23 


928181 


1.87 


818899 
814175 
814452 


4-60 


186101 


55 


6 1 


737274 


3-23 


928098 


1.87 


4-6o 


185825 


54 , 


2 


737467 


3-23 


923016 


1.87 


4.60 


185548 


58 1 


737661 


3-22 


922933 


1.37 


814728 


4.60 


185272 


52 


9 


737855 
738048 


3-22 


92205l 


III 


8i5oo4 


4.60 


184996 


5i 


iO 


3-22 


922768 


815279 


4-60 


184721 


5o 


II 


9-738241 


3-22 


9.922686 


1.38 


9.815555 


4.59 


io.i8/j/|/i5 


it 


i« 


738434 


3-22 


922603 


1.88 


81 588 1 


4.59 


184169 
188898 


i3 


738627 


8.21 


922520 


1.88 


816107 


4-59 


47 


14 


738820 


3.21 


922438 


1-88 


816882 


4.59 


188618 


46 


i5 


739013 


3.21 


922355 


1.88 


8i6658 


4.59 


188842 


45 


i6 


739206 


3.21 


922272 


1.88 


816988 


4.59 


188067 


44 


\l 


739398 


3.21 


922189 


1.88 


817209 


4.59 


182791 


43 


739500 


3*20 


922106 


1.88 


817484 


4.59 


i825i6 


42 


19 


739783 


3.20 


922028 


1.88 


817759 
818085 


4-5o 


182241 


41 


20 


739975 


3-20 


921940 


1.88 


4-58 


181965 


40 


21 


9.740167 


3.20 


9.921857 


1.39 


9.818810 


4-58 


10.181690 


39 


22 


-40359 


3.20 


921774 


1.89 


8i8585 


4.58 


i8i4i5 


88 


23 


74o55o 


3.19 


921691 


1.89 


818860 


4-58 


181140 


37 


24 


740742 


3.19 


921607 


1.39 


819185 


4.58 


i8o865 


36 


25 


740934 


3.19 


921524 


1.89 


819410 


4.58 


iSoSgo 


35 


26 


741 125 


3-19 


921441 


1.89 


819684 


4.58 


180816 


34 


11 


74i3i6 


3-19 
3.18 


921357 


1.89 


819959 


4.58 


180041 


33 


741 5o8 


921274 


1.39 


820284 


4.58 


179766 


8£ 


29 


741699 


3.18 


921190 


1.89 


82o5o8 


4.57 


179492 


3i 


3o 


741889 


3.18 


921107 


1.89 


820788 


4.57 


179217 


3o 


3i 


9 -742080 


3-18 


9.921023 


1.89 


9-821057 


4.57 


10.178943 


29 


32 


742271 


3.18 


920989 


1.40 


821882 


4.57 


178668 


28 


33 


742462 


3.17 


920856 


1.40 


821606 


4.57 


178894 


27 


34 


742652 


3-17 


920772 


1.40 


821880 


4-57 


178120 


26 


35 


742842 


3.17 


920688 


1.40 


822154 


4.57 


177846 


25 


36 


743o33 


3-17 


920604 


1.40 


822429 


4.57 


177571 


24 


11 


743223 


3.17 


920520 


1.40 


822700 


4.57 


177297 


23 


743413 


3-16 


920486 


1.40 


822977 


4-56 


177028 


22 


39 


743602 


3-16 


920352 


1.40 


828250 


4.56 


176750 


21 


40 


743792 


3-i6 


920268 


1.40 


828524 


4-56 


176476 


20 


41 


9-743982 


3.16 


9-920184 


1. 40 


9.828798 


4.56 


10.176202 


\t 


42 


744171 


3.16 


920099 


1.40 


824072 


4-56 


175928 


43 


744361 


3-i5 


920016 


1.40 


824845 


4-56 


175655 


\l 


44 


744550 


3-i5 


919981 


1.41 


824619 
824898 


4-56 


175881 


45 


744730 
744928 


3.i5 


919846 


1.41 


4-56 


175107 


i5 


46 


3-i5 


919762 


1.41 


825166 


4.56 


174884 


14 


% 


745117 
745306 


3-i5 


919677 


1.41 


825489 
825713 


4.55 


174561 


i3 


3.14 


919598 


1.41 


4.55 


174287 


12 


49 


745494 


3.14 


919508 


1.41 


825986 


4.55 


174014 


II 


5o 


745683 


3-14 


919424 


1. 41 


826259 


4-55 


173741 


10 


5i 


9.745871 
746o5o 


3-14 


9.919889 


1.41 


9.826532 


4-55 


10.173468 


Q 


52 


3.14 


919254 


1. 41 


826805 


4-55 


178195 





53 


746240 


3-13 


919169 


1. 41 


827078 


4-55 


172922 


I 


54 


746436 


4-13 


919085 


1.41 


827851 


4.55 


172649 


55 


746624 


3.i3 


919000 


1. 41 


827624 


4.55 


172376 


5 


56 


746812 


3-13 


910915 


1.42 


827897 
828170 


4.54 


172108 


4 


U 


746909 
747 « 87 


3.i3 


918880 


1.42 


4.54 


171880 


3 


3.ii 


918745 


1.42 


828442 


4-54 


171558 


a 


59 


747374 


3-12 


918659 


1.42 


828715 


4.54 


171285 


1 


60 


747562 


3.12 


918574 


1.42 


828987 


4-54 


171013 





Coeine 


D. 


Sine 


D. 


Cotang. 


D. 


Tang. 



18 



(56 DEGREES.) 



52 



C34 DEGREES.) A TABLE OF LOGARITHMIC 



M. 




Sine 


D. 


Cosine 


D. 


, Tang. 


D. 


1 Cotang. 




9-747562 


3-12 


9-918574 


1.42 


9-828987 


4.54 


10-171013 


60 


I 


747749 


3-12 


' 918489 


1-42 


829260 


4-54 


I 170740 5o 
170468 58 


a 


747936 


3-12 


918404 


! 1.42 


829532 


4-54 


3 


748123 


3-11 


i 9i83i8 


i 1-42 


829805 


4-54 


170195 57 


4 


748310 


3-11 


j 918233 


1-42 


83oo77 


4-54 


169923 56 


5 


748497 


3-II 


918147 


1-42 


83o349 


4-53 


169651 


55 


6 


748683 


3ii 


1 918062 


1-42 


83o62i 


4-53 


169379 


54 


I 


748870 


3-11 


917976 


1-43 


830893 


4-53 


169107 

168835 


53 


749036 


3-10 


i 917891 


1-43 


83ii65 


4-53 


52 


9 


749243 


3-10 


I 917805 


1-43 


831437 


4-53 


168563 


5i 


10 


749429 


3-10 


917719 


1-43 


831709 


4-53 


168291 


5o 


II 


9-749615 


3-10 


9-917634 


1-43 


9-831981 


4-53 


10-168019 


S 


12 


749801 


3-10 


917548 


1-43 


832253 


4-53 


167747 
167475 


i3 


749987 


3-09 


917462 


1-43 


832325 


4-53 


2 


14 


750172 


3-09 


917376 


1-43 


832796 


4-53 


167204 


i5 


75o358 


3-09 


917290 


1-43 


833o68 


4-52 


166932 


45 


i6 


700543 


3.09 


917204 


1-43 


833339 


4-52 


166661 


44 


\l 


750729 


3-oc 


917118 


1-44 


833611 


4-52 


166389 
166118 


43 


750914 


3.08 


917032 


1.44 


833882 


4-52 


42 


19 


751099 


3-o8 


916946 


1-44 


8341 54 


4-52 


165846 


41 


20 


751284 


3-08 


916059 


1-44 


834425 


4-52 


165075 


40 


21 


9-751469 


3 -08 


9-916773 


1-44 


9-834696 


4-52 


io-i653o4 


1? 


22 


75 I 654 


3-o8 


916687 


1-44 


834967 
835238 


4-52 


i6oo33 


23 


75i83q 


3-o8 


916600 


1-44 


4-52 


164762 


u 


24 


75202J 


3-07 


9i63i4 


1-44 


835309 


4-52 


164491 


25 


732208 


3-07 


916427 


1-44 


830780 


4-5i 


164220 


35 


26 


752392 


3-07 


916341 


1-44 


836o5i 


4-5i 


163949 


34 


11 


732376 


3.07 


916234 


1-44 


836322 


4-5i 


163678 


33 


752760 


3.07 


916167 


1-45 


836593 


4-5i 


163407 
i63i36 


33 


29 


752944 


3.06 


916081 


1-45 


836864 


4-5i 


3i 


3o 


753128 


3.06 


915994 


1-45 


837134 


4-5i 


162866 


3o 


3i 


9.753312 


3-o6 


9-913907 


1-45 


9-837405 


4-5i 


10-162595 


29 
28 


32 


•753495 


3 -06 


913820 


1-45 


837675 


4-5i 


162325 


33 


753679 


3-06 


913733 


1-45 


837946 


4-5i 


162054 


27 


34 


753862 


3-o5 


915646 


1-45 


838216 


4-5i 


161784 


26 


35 


754046 


3-o5 


915559 


1-45 


838487 


4-5o 


i6ioi3 25 


36 


734229 


3-o5 


915472 


1-45 


838757 


4-5o 


161243 24 


u 


754412 


3-o5 


915385 


1-45 


83go27 


4-5o 


160973 23 i 


•754595 


3 -05 


915297 


1-45 


839297 
839568 


4-5o 


160703 22 


39 


754778 


3-04 


915210 


1-45 


4-5o 


160432 21 


40 


754960 


3-04 


9i5i23 


1.46; 


839838 


4-5o 


160162 20 


41 


9-755143 


3-04 


9-9i5o35 


1-46 1 


9-840108 


4-5o 


10-159892 19 
109622 18 


42 


755326 


3-04 


914948 


1-46 1 


840378 


4-5o 


43 


755508 


3-04 


914860 


1-46 


840647 


4-5o 


159353 17 
159083 16 ' 


44 


755690 


3-04 


914773 1 


1-46 


840917 


4-49 


45 


755872 


3-o3 


914683 1 


1.46 


841187 


4.49 


i588i3 i5 ■ 


46 


756o54 


3 -03 


914598 1 


1-46 


841457 


4.49 


158543 14 


S 


756236 


3-03 ' 


914010 ; 


1-46 1 


841726 


4-49 


158274 i3 


756418 


3-03 


914422 j 


1-46 


841996 1 


4-49 1 


i58oo4 12 


49 


756600 


3-o3 


914334 1 


1-46 


842266 i 


4-49 ; 


157734 II 


5o 


706782 


3-02 


914246 


1-47 


842535 1 


4.49 


157465 10 


5i 


9-756963 


3-02 


9-914158 j 


1-47 


9 842805 i 


4.49 


10-157195 
156926 8 


55 


757144 


3-02 


914070 i 


1-47 


843074 i 


4-49 1 


53 


757326 


3-02 ■ 


913982 i 


1-47 


843343 1 


4.49 1 


I 56657 7 ' 


54 


757507 


3-05 


913894 i 


1-47 


843612 ' 


4-4Q . 
4-48 


156388 6 


55 


757688 


3-01 


9i38o6 


1-47 


843882 1 


i56ii8 i 5 


56 


757869 


3-01 


913718 


1-47 1 


844i5i 1 


4-48 


155849 4 


ll 


758o5o 


3-01 


9i363o 1 


1-47 


8/i/i/i20 


4-48 


i5558o ' 3 


758230 


3*01 


913541 


1-47 


844689 


4-48 I 


i553ii 


3 


59 


75841 I 


3*01 


913453 


1-47 


844958 : 


4-48 


i55o42 


I 


6o 


753591 


3*01 


913365 


1-47 


845227 ; 


4.48 


154773 







Coeine 


D. 


Sine 


D. ! 


Cotang. 1 


D. ! 


Tang. ' M. ; 








(55 


DSGRS 


,S8.) 









SINttS AND f AISTQENTS. (35 DEGREES.; 



53 



M. 


Sino 


D. 


Coftine 


D. 


Tang. 


D. 


Cotang. 


60 





o-7585oi 


3'0i 


9.913365 


1-47 


9.845227 
845496 


4-48 


10.154773 
1 54504 


I 


fslll 


3 


00 


918276 
9181S7 


1-47 


4.48 


58 


1 


3 


00 


1-48 


845764 


4.48 


154236 


3 


759132 


3 


00 


918099 


1-48 


846033 


4.48 


158967 
1 53698 
i5848o 


U 


4 


759312 


3 


00 


918010 


1.48 


846802 


4-48 


5 


759492 


3 


00 


912922 


1.48 


846570 


4-47 


55 


6 


759672 
759852 


2 


99 


912883 


1-48 


846889 


4-47 


i53i6i 


54 


I 


2 


99 


91 51744 
912655 


1.48 


847107 
847876 


4-47 


152898 


53 


760031 


2 


99 


1.48 


4-47 


152624 


52 


9 


7602 1 1 


2 


99 


912566 


1.48 


847644 


4-47 


152356 


5i 


10 


760890 


2 


99 


912477 


1.48 


847918 


4-47 


152087 


5o 


II 


Q- 760560 


2 


98 


9.912888 


1.48 


9-848181 


4-47 


io-i5i8i9 


it 


13 


760748 


2- 


98 


912299 


1-49 


848449 


4-47 


i5i55i 


i3 


760927 


2 


98 


912210 


1-49 


848717 
848986 


4-47 


i5i283 


47 


14 


761 106 


2- 


98 


912121 


1-49 


4-47 


i5ioi4 


46 


i5 


761285 


2- 


98 


912081 


1-49 


849254 


4-47 


150746 


45 


i6 


761464 


2- 


98 


91 1942 


1-49 


849522 


4-47 
4.46 


150478 


44 


\l 


761642 


2" 


97 


9u858 


1-49 


849790 


l5o210 


43 


761821 


2- 


97 


911763 


1-49 


85oo58 


4.46 


149943 


43 


«9 


761999 


2< 


97 


91 1674 
911 584 


1-49 


85o825 


4.46 


149675 


41 


20 


762177 


2- 


97 


1-49 


85o593 


4.46 


149407 


40 


21 


9-762356 


2- 


ll 


9-9II495 


1-49 


9 -850861 


4.46 


10-149139 


It 


22 


762534 


2- 


911405 


1-49 


851129 


4.46 


14887 1 


33 


762712 
762889 


2. 


96 


9ii3i5 


i.5o 


851896 


4.46 


148604 


ti 


24 


2. 


96 


911226 


i.5o 


85 1664 


4.46 


148386 


25 


768067 


2- 


96 


911186 


1-50 


851981 


4.46 


148069 


35 


36 


768245 


2- 


96 


91 1046 


i-5o 


852199 


4-46 


I 4780 I 


34 


11 


768422 


2- 


96 


910956 


i.5o 


852466 


4.46 


147534 


33 


768600 


2- 


95 


910866 


I -50 


852788 


4.45 


147267 


32 


29 


768777 


2- 


95 


910776 


i.5o 


853001 


4.45 


146999 
146733 


3i 


3o 


768954 


2- 


95 


910686 


i.5o 


853268 


4-45 


3o 


3i 


9.764181 


2 


95 


9.910596 


i.5o 


9-853585 


4.45 


10-146465 


20 


32 


764808 


2. 


95 


9io5o6 


i.5o 


858802 


4-45 


146108 
1459 J I 


2d 


33 


764485 


2- 


94 


9io4i5 


i.5o 


854069 


4.45 


u 


34 


764662 


2- 


94 


910825 


i.5i 


854386 


4.45 


145664 


35 


764888 


2- 


94 


910235 


iSi 


854603 


4.45 


145397 
i45i3o 


25 


36 


76501 5 


2 


94 


910144 


iSi 


854870 


4.45 


34 


ll 


765191 


2 


94 


910054 


1-51 


855187 


4.45 


144868 


23 


760867 


2 


94 


909968 


i.5i 


855404 


4.45 


144596 


22 


h 


765544 


2 


93 


909878 
909782 


1-51 


855671 


4-44 


144829 


21 


4o 


765720 


2 


93 


1-51 


855938 


4.44 


144062 


20 


4i 


9.765896 


2 


93 


9-909691 


1-51 


9-856204 


4.44 


10-143796 


\t 


42 


766072 


2 


93 


909601 


i.5i 


856471 


4-44 


143520 
148268 
142996 
142780 


43 
44 


766247 
760428 


2 

2 


93 

93 


909510 
909419 


i-5i 
i.5i 


856787 
857004 


4-44 
4-44 


]l 


45 


766598 


2 


92 


909828 


1-52 


857270 


4-44 


i5 


46 


766774 


2 


92 


909287 


1-52 


857537 


4-44 


142463 


14 


48 


766949 


2 


92 


909 1 46 


1-52 


857808 


4.44 


142197 
141901 


i3 


767124 


3 


92 


909055 


1-52 


858069 


4-44 


12 


i 49 


767800 


2 


92 


908064 


1-52 


858836 


4.44 


141664 


II 


5c 


767475 


2 


91 


908878 


1-52 


858602 


4.43 


141398 


10 


5i 


9 • 767649 


2 


9« 


9-908781 


1.52 


9-858868 


4-43 


io-i4ii8a 


t 


52 


767824 


3 


91 


908690 


1-52 


859184 


4-43 


140866 


53 


tivn 


2 


91 


908599 


1-52 


859400 


4-43 


140600 


I 


1 54 


2 


91 


9o85oT 
908410 


1-52 


859666 


4-43 


140884 


55 


768348 


3 


90 


1-53 


859982 


4-43 


140068 


5 


56 


768522 


2 


90 


908824 


1.53 


860198 


4-43 


139802 


4 


u 


768697 


2 


90 


908233 


1.53 


860464 


4.43 


189536 


3 


768871 


3 


■ 90 


908141 


1.53 


860780 


4-43 


189270 


3 


59 


769045 


2 


.90 


908049 


1.53 


860995 


4-43 


189005 
188739 


I 


60 


769219 


3.90 


907958 


1.53 


861261 


4-43 







Cosine 


D. 


Sine 


D. 


Cotang. 


D. 


Tang. 


M 

—^-j 



(64 DEGREES.) 



54 



(36 DEGREES.) A TABLE OF LOGARITHMIC 



"m. 


Sine 


D 


1 Cosine D. 


Tang. 


D. 


Cotang. 


— -J 





9.769310 
769393 


a -90 


I 9-907958 I 
i 907866 I 


.53 


' 9-861261 


4.43 


10-138739 


60 


I 


2 


.89 


.53 


j 86i527 


4 


•43 


138473 1 59 


3 


769566 


2 


89 


907774 I 


-53 


861792 
862058 


4 


•4a 


' i38ao8 1 58 


3 


769740 


a 


.89 


907682 I 


-53 


4 


-42 


13794a 1 57 
137677 1 56 


4 


769913 


a 


.89 


907590 I 


-53 


862828 


4 


•42 


5 


770087 


2 


.89 


' 907498 I 


53 


862389 


4 


42 


187411 55 


6 


770260 


a 


88 


907406 I 


53 


862854 


4 


•42 


1 87 146 1 54 


2 


770433 


2 


88 


■ 907314 I 


54 


868119 


4 


-42 


i3688i 


53 


770606 


2 


88 


907222 I 


54 


868385 


4 


42 


i366i5 


5a 


9 


770779 


2 


88 


907129 . I 


54 


868650 


4 


•42 


I 36350 


5i 


10 


77095a 


2 


88 


' 907037 ■ I 


54 


8639x5 


4 


•42 


i36o85 


5o 


II 


9.771125 


2 


88 


9-906945 I 


54 


' 9-864180 


4 


42 


10-135820 


S 


la 


771298 


2 


87 


9o6o52 1 


54 


864445 


4 


42 


135555 


i3 


771470 


2 


87 


906760 I 


54 


864710 


4 


42 


135290 


ii 


14 


771643 


2 


87 


906667 I 


54 


864975 


'4 


41 


i35o25 


i5 


771815 


2 


87 


906575 I 


54 


865240 


4 


41 


184760 


45 


i6 


771987 


2 


87 


906482 I 


54 


8653o5 


4 


41 


184495 


44 


\l 


772159 


2 


87 


906889 I 


55 


865770 


4 


4i 


134280 


43 1 


772331 


2 


86 


906296 I 


55 


866o35 


4 


41 


188965 42 


19 


7725o3 


2 


86 


906204 I 


55 


866800 


4 


41 


133700 I 41 i 


ao 


772675 


a 


86 


9061 1 1 I 


55 


866564 


4 


41 


133436 


40 


31 


9-772847 
773018 


2 


86 


9-906018 I 


55 


9-866829 


4 


41 


10.133171 


39 


aa 


2 


86 


905925 I 


55 


867094 
867358 


4 


41 


182906 i 38 i 


a3 


773190 


2 


86 


9o583a . i 


55 


4 


41 


182642 


u 


24 


773361 


2 


85 


905739 I 


55 


867628 


4 


41 


182877 


a5 


773533 


2 


85 


903643 I 


55 


867887 


4 


4i 


182113 


35 


26 


773704 


a 


85 


9o555a i 


55 


868i52 


4 


40 


181848 


34 


11 


773875 


2- 


85 


905459 I 


55 


868416 


4 


40 


i3i584 


33 


774046 


2 


85 


9o5366 I 


56 


868680 


4 


40 


i3i320 


33 


29 


774217 


2 


85 


905272 I 


56 


868945 


4 


40 


i3io55 


3i 


3o 


774388 


2 


84 


905179 I 


56 


869209 


4 


40 


180794 


3o 


3i 


9-774558 


2 


84 


9 -905085 I 


56 


9-869473 


4 


40 


io.i3o527 


*S 


32 


774729 
774899 


2 


84 


904992 I 
904898 I 


56 


869787 


4 


40 


180268 


ad 


33 


2 


84 


56 


870001 


4 


40 


129999 
129735 


27 


34 


775070 


2- 


84 


904804 i I 


56 


870265 


4 


40 


26 


35 


775240 


2- 


84 


9047" ! I- 


56 


870529 


4 


40 


129471 


25 


36 


775410 


2- 


83 


904617 ! I- 


56 


870793 


4 


40 


139207 


24 


ll 


775580 


2- 


83 


904523 j I- 


56 


871057 


4 


40 


128943 


23 


775750 


2- 


83 


904429 I- 
904335 1 I- 


57 


871821 


4 


40 


128679 
128415 


23 


39 


775920 


2- 


83 


57 


871585 1 


4 


40 


21 


40 


776090 


2- 


83 


904241 i I- 


57 


871849 


4- 


39 


ia8i5i 


20 


41 


9-776259 


2- 


83 


9-904147 i I- 


57 


9-872112 


4- 


39 


10-127888 


:? 


4a 


776420 


2- 


82 


904053 1 I' 


57 


872876 


4 


39 


127634 


43 


77659§ 


2- 


82 


908959 1 I' 


^7 


872640 


4 


39 


127860 I 17 1 


44 


776768 


2- 


82 


908864 i I- 


57 


872908 


4- 


39 


127097 1 16 1 


45 


776937 


2- 


82 


908770 ' I. 


57 


878167 


4 


39 


126833 


i5 


46 


777106 


2- 


82 


908676 I- 


57 


878480 


4 


39 


126570 


14 


tl 


777275 


2- 


81 


9o858i I- 


57 


878694 


4 


39 


126806 


i3 


777444 


2- 


81 


908487 I- 


57 


878957 


4- 


39 


126043 j 12 1 


49 


777613 


2- 


81 


908892 I- 


58 


874220 : 


4- 


39 


125780 11 1 


50 


777781 


3' 


81 


908298 I- 


58 


874484 


4- 


39 


i255i6 


10 


5i 


9.777950 


2- 


81 


9-908208 I- 


58 


9-874747 


4- 


39 


ioi25a53 


t 


5? 


7781 19 


2' 


81 


908108 I- 


58 


875010 ! 


4- 


It 


124990 


53 


778287 


2- 


80 


908014 I- 


58 


875273 ' 


4- 


134727 


I 


54 


778455 


2- 


80 


902919 ! I- 


58 


875536 


4- 


38 


134464 


55 


778624 


2- 


80 


902824 1 I- 


58 


875800 ; 


4- 


38 


134300 


5 1 


56 


778792 


2- 


80 


902729 ! I- 


58 


876063 j 


4- 


38 


133987 


4 


u 


778960 


2- 


80 


902634 ! I- 


58 


876826 1 


4 


38 


138674 


3 


7-79128 


a- 


80 


902539 I- 


59 


876589 , 


4- 


38 


133411 


3 


59 


779295 


a- 


■'9 


902444 I- 


59 


876851 


4- 


38 


ia3i49 


I 


60 


779463 


2-79 


902349 1 I- 


59 


8771U 


4-38 


133886 







CkMsme 


D. 


Sine i D. 


Cotang. 1 


D. 


Tang. 


'^L^ 



(53 DBGRESS.) 



SINES AND TANGENTS. (37 DEGREES.; 



5d 



M. 

o 


Sine 


D. 


Cosino I 


>. 


Tang. 


D. 


Cotang. 




Q- 779463 


2-79 


9-902840 1 

90225j I 


59 


9-877"4 


4-38 


10-122886 


60 


I 


779631 


2- 


79 


59 


877377 


4- 


88 


122623 


i? 


a 


779798 


2- 


79 


902158 1 


59 


877640 


4 


88 


122860 


3 


780133 


2- 


79 


902068 I 


^9 


877908 


4 


88 


122097 


57 


4 


2- 


"79 


901967 I 


59 


878165 


4 


88 


121835 56 1 


5 


780300 


2- 


78 


901872 I 


?9 


878428 


4 


88 


121572 


55 


6 


780467 


2- 


78 


901776 I 


59 


878691 


4 


88 


121809 


54 


I 


780634 


2- 


7^ 


901681 1- 


59 


878953 


4 


37 


121047 


53 


780801 


2- 


78 


901 585 I- 


59 


879216 


4 


37 


120784 

120322 


5a 


9 


780968 


2- 


7? 


901490 I 


59 


879478 


4- 


37 


5i 


10 


781134 


2< 


78 


901394 I 


60 


879741 


4- 


37 


120259 


5o 


II 


9.781301 


2< 


77 


9-901298 1 


60 


9-880008 


4- 


37 


10-119997 


49 


12 


781468 


2 


77 


901202 1 


60 


880265 


4 


37 


119785 


48 


i3 


781634 


2< 


77 


901 106 I 


60 


880528 


4 


37 


119472 


47 


14 


781800 


2< 


77 


OOIOIO I 


60 


880790 


4 


37 


119210 


46 


i5 


781966 


2- 


77 


900914 I 


60 


88io52 


4 


37 


118948 


45 


i6 


782182 


2- 


77 


900018 I 


60 


88i3i4 


4- 


37 


1 1 8686 


44 


\l 


782298 


2- 


76 


900722 I 


60 


881576 


4 


37 


118424 


43 


782464 


2- 


7^ 


900626 1 1 


60 


881889 


4 


37 


118161 


42 


»9 


782630 


2- 


76 


900529 I 
900433 1 ■ 


60 


882101 


4 


37 


117899 


41 


20 


782796 


2- 


76 


61 


882868 


4 


36 


117687 


40 


21 


9.782961 


2- 


76 


9-900387 I- 


61 


9-882625 


4 


36 


10-117875 


39 


22 


788127 


2- 


76 


900240 1 • 


61 


882887 


4 


86 


117118 


38 


23 


788292 
788458 


2- 


75 


900144 1 


61 


888x48 


4 


86 


116852 


37 


24- 


2- 


75 


900047 1 • 


61 


883410 


4 


36 


116590 


36 


25 


788628 


2- 


75 


899951 1 


61 


888672 


4 


86 


116828 


35 


26 


788788 


2- 


75 


899854 I • 


61 


888984 


4 


36 


1 16066 


34 


\l 


788953 


2- 


75 


899757 I 


61 


884196 


4 


36 


ii58o4 


33 


784118 


2- 


75 


899660 I 


61 


884457 


4 


86 


115543 


32 


29 


784282 


2- 


74 


899564 1 ■ 


61 


884719 


4 


36 


ii528i 


3i 


3o 


784447 


2- 


74 


899467 1 


62 


884980 


4 


36 


Il5020 


3o 


3i 


9-784612 


2- 


74 


9-899870 1 


62 


9-885242 


4 


86 


10-114758 


^ 


32 


784776 


2 


74 


899278 1 


62 


8855o8 


4 


86 


1 14497 
114235 


33 


784941 


2 


74 


899176 1 


62 


885765 


4 


36 


27 


34 


785io5 


2 


74 


899078 1 


62 


886026 


4 


86 


118974 


26 


35 


785269 
785488 


2 


73 


898981 1 


62 


886288 


4 


86 


118712 


25 


36 


2< 


73 


898884 I 


62 


886549 


4 


85 


ii345i 


24 


11 


785597 


2 


73 


898787 1 


62 


886810 


4 


85 


118190 


23 


785761 


2 


73 


898689 I 


62 


887072 


4 


85 


112928 


22 


39 


785925 


2 


73 


898592 I 


62 


887888 


4 


35 


112667 


21 


40 


786089 


2 


73 


898494 I 


63 


887594 


4 


85 


1 1 2406 


20 


41 


9-786252 


2 


72 


9-898897 I 


68 


9-887855 


4 


35 


10-112145 


19 

18 


42 


786416 


2 


72 


898299 I 


68 


8881 16 


4 


35 


111884 


45 


786579 


2 


72 


898202 1 


68 


888877 


4 


35 


111623 


17 


U 


786742 


2 


72 


898 I Od 1 


68 


888689 


4 


35 


111861 


16 


45 


786906 


2 


72 


898006 1 


68 


888900 


4 


35 


11 1 100 


i5 


46 


787069 


2 


72 


897908 1 


68 


889160 


4 


35 


110840 


14 


47 


787282 


2 


71 


897810 1 


68 


889421 


4 


35 


110579 


i3 


48 


787895 


2 


71 


897712 1 


68 


889682 


4 


35 


110810 


la 


49 


787557 


2 


71 


897614 1 


68 


889943 


4 


35 


110057 


II 


5o 


787720 


2 


V 


897516 I 


63 


890204 


4 


34 


•09796 


10 


5i 


9-787888 
788045 


2 


71 


9-897418 1 


64 


9 • 890465 


4 


34 


ro- 109535 


t 


52 


2 


71 


897820 I 


64 


890725 


4 


34 


109275 


53 


788208 


2 


71 


897222 I 


64 


890986 


4 


34 


109014 


7 


54 


788870 
788532 


2 


70 


897128 1 


64 


891247 


4 


34 


108753 


6 


55 


2 


70 


897025 1 


64 


891507 


4 


34 


108498 
108282 


5 


56 


788694 
788856 


2 


70 


896926 I 


64 


891768 


4 


34 


4 


5i 


2 


.70 


896828 1 


-64 


892028 


4 


34 


107972 


3 


5o 


789018 


2 


.70 


896729 1 


64 


892289 


4 


34 


1077 I I 


a 


59 


789180 


a 


.70 


896681 I 


64 


892549 


4 


.34 


107451 


I 


60 


789342 


a-69 


896582 1 


64 


892810 


4.34 


107190 





Goaine 


D. 


Sine D. 


CoUng. 


D. 


Tang. 


M. 



(52 DSORKI8.) 



56 



(38 DEGRKES.) A TABLE OF LOGARltmttO 






^ Sine 


D. 


Cosine ; D. 


Tang. 


D. 


Cotwig. 




9-789343 


3.69 


9.896532 1 


-64 


9-892810 


4-34 


10.107190 


60 


I 


789504 


3 


69 


896433 ! 1 


• 65 


893070 


4.34 


106930 


5o 


3 


789665 


2 


^ 


896335 1 1 


.65 


893331 


4.34 


106669 


59 


3 


789827 


3 


.69 


896236 1 I 


-65 


893591 


4.34 


106409 


67 


i 


789988 


2 


69 


896137 1 1 
89603d 1 


• 65 


893801 


4 34 


1 06 149 


56 


5 


790149 


2 


^ 


•65 


8941 1 1 


4-34 


106889 i 55 


6 


7903 10 


2 


68 


895939 , 1 


• 65 


894371 


4.34 


106629 I 54 
io536? I 53 1 


i I 


790471 


2 


• 68 


895840 1 


• 65 


894632 


4-33 


790632 


2 


• 68 


893741 I 


-65 


894892 
893152 


4-33 


10610& 


53 


i 9 


790793 
790954 


2 


.68 


895641 1 


• 65 


4-33 


104848 


5i 


! ic 


2 


.68 


895542 1 


•65 


895412 


4-33 


104688 


5o 


II 


9.791115 


3 


68 


9.895443 I 


-66 


9-895672 


4.33 


10.104328 


49 


12 


791275 


3 


67 


895343 I 


-66 


895932 


4-33 


104068 


48 


i3 


791436 


2 


67 


895244 1 


• 66 


896192 


4.33 


io38o8 


47 


14 


791596 


2 


.67 


895145 I 


• 66 


896432 


4.33 


103648 


46 


i5 


791757 


2 


.67 


895045 1 


• 66 


896712 


4.33 


103288 


45 


i6 


791917 


3 


.67 


894045 1 I 


-66 


89697 1 


4-33 


io3o29 


44 


17 


792077 


3 


67 


894846 i 1 


• 66 


897231 


4.33 


102769 


43 


i8 


792237 


2 


66 


894746 I 


• 66 


897491 


4-33 


102609 


43 


19 


792397 


2 


66 


894646 1 


• 66 


897751 


4-33 


102249 


41 


20 


792557 


2 


66 


894546 1 


.66 


898010 


4.33 


101990 


40 


21 


9.792716 
792876 


2 


66 


9-894446 1 


• 67 


9-898270 


4-33 


10.101730 


^ 


22 


2 


66 


894346 1 


• 67 


898530 


4.33 


101470 


23 


793o35 


2 


66 


894246 I 


.67 


898789 


4.33 


101211 


37 


24 


793iq5 


2 


65 


894146 1 


67 


899049 


4.32 


100961 


36 


25 


793354 


2 


65 


894046 1 


67 


899308 


4-32 


100692 
100432 


35 


26 


793514 


2 


65 


893946 1 


-67 


899568 


4-32 


34 


U 


793673 


3 


65 


893846 1 


67 


899827 


4.32 


100173 


33 


793832 


2 


65 


893745 1 


67 


900086 


4.32 


099914 


32 


29 


793991 


2 


65 


893645 I 


67 


900346 


4.32 


099664 


3i 


3o 


7941 5o 


2 


64 


893544 1 


67 


900605 


4-32 


099396 


3o 


''' 3i 


9.794308 


2 


64 


9-893444 I 


68 


9 • 900864 


4.32 


io^o99i36 
090876 


39 


' 32 


794467 


2 


64 


893343 I 


68 


901 1 24 


4.32 


38 


33 


794626 


2 


64 


893243 I 


68 


90i383 


4.32 


098617 
098368 


11 


; 34 


794784 


2 


64 


893142 I 


68 


901642 


4-32 


35 


794942 


2 


64 


893041 I 


68 


901901 


4.32 


098099 


25 


36 


795^01 


3 


64 


892940 I 


68 


902160 


4.32 


097840 


24 


u 


795259 


3 


63 


892839 I 


68 


902419 


4.32 


097681 


33 ; 


795417 


3 


63 


892739 I 


68 


902679 


4.32 


097321 


33 i 


39 


795575 


3 


63 


892638 I 


68 


902938 


4.33 


097062 


31 1 


40 


795733 


3 


63 


892536 I 


68 


903197 


4-3i 


096803 


30 1 


41 


9.795891 


3 


63 


9-892435 I 


69 


9-903455 


4.31 


10 •096646 


\i 


42 


796049 


2 


63 


892334 I 


69 


903714 


4.31 


096286 


43 


796206 


2 


63 


892233 1 


69 


903973 


4.31 


096027 
096768 


\i\ 


44 


796364 


2 


62 


892132 ^ I 


69 


904232 


4.31 


45 


796521 


2 


62 


892030 1 I 


69 


904491 


4.31 


096609 


16 ! 


46 


796679 


2 


62 


891029 1 I 


69 


904730 


4-31 


096260 


14 ' 


s 


796836 


2 


62 


801827 I 


69 


903008 


4-31 


094993 
094733 


i3 1 


796993 
797 I 5o 


2 


62 


89*1726 I 


69 


906267 


4-31 


12 1 


49 


3 


61 


891624 1 


69 


906626 


4.31 


094474 


11 1 


5o 


797307 


3< 


61 


891523 I 


70 


906784 


4.31 


094216 


10 j 


i' 


9-797464 


2- 


61 


9.891421 1 


70 


9 - 906043 


4-31 


10.093967 
093698 


?i 


53 


797621 


3 


6i 


891319 I 


70 


9o63o2 


4.31 


53 


797777 


3 


61 


891217 1 


70 


906660 


4-31 


093440 


7 


54 


797934 
798091 


3 


61 


891115 I 


70 


90681Q 


4-31 


093181 


6 ' 


55 


2 


61 


891013 I- 


70 


907077 


4-31 


092923 


5 i 


5e 


798247 


3 


61 


89091 1 I . 


70 


907336 


4-31 


092664 


4 


U 


798403 


2 


60 


890809 1 • 


70 


907694 


4.31 


092406 


3 


798660 


2- 


60 


890707 1 I 


70 


907862 
908111 


4.31 


092148 3 


59 


798716 


3" 


60 


8oo6o5 1 1- 


70 


4-30 


091889 1 


60 


798871 


3.60 


890603 1 I 


70 


908369 


4.30 


O9163I 




CoBine 


D. 


Sine i r 


). 


Cotang. 1 


D. 


Taxifr* 1 ^* 



(51 DEGREES.) 





SINKS AND TANGEl 


^S 


(39 DEGREES 


*J 


&? 


M." 


] Sine 


D. 


Cosine D. 


Tang. 


D. 


CotAng. 


60 





9.798872 


2'6o 


9.890503 I 


.70 


9.908369 


4-3o 


10-0191631 


I 


799028 


2-60 


890400 j 1 


•71 


908628 


4.30 


091372 


U 


7 


799184 


2 -60 


890198 1 


•71 


908886 


j 4-3o 


091114 


3 


799339 


2-59 


890195 1 


•71 


909144 


4 -30 


090856 


ll 


4 


799493 


2-59 


, 890093 ; I 


•71 


'' 909402 


4-3o 


1 090398 


56 


5 


799651 


2-59 


889990 i 1 


•71 


909660 


4-3o 


1 090340 


55 


6 


799806 


2-59 


889888 ! I 


•71 


909918 


4-3o 


39008s : 54 ' 


I 


799962 


2-59 


889785 ' I 


•71 


910177 


: 4-3o 


089823 ' 53 


8001 17 


2-59 


889682 . I 


•71 


910435 


4-3o 


089565 


52 


9 


800272 


2-58 


889579 i 1 


•71 


910693 


4-3o 


089307 


5i 


10 


800427 


2-58 


889477 i I 


•71 


910951 


4-3o 


089049 


5o 


\i 


9800582 


2-58 


9.889374 ' I 


.72 


9-911209 


4-3o 


10-088791 


49 


12 


800737 
800092 


2.58 


889271 1 I 


• 72 


91 1467 


4-3o 


088533 


48 


i3 


2-58 


889168 ' I 


-72 


911724 


4-3o 


088276 


47 


14 


801047 


2.58 


889064 i 1 


.72 


91 1982 


4-3o 


088018 


46 


i5 


801201 


2.58 


888961 ; I 


-72 


912240 


4-3o 


087760 


45 


i6 


8oi356 


2.57 


888858 , I 


.72 


912498 


4-3o 


087502 


44 


\l 


8oi5ii 


2.57 


888755 I I 


72 


9127D6 


4-3o 


087244 


43 


80 I 665 


2.57 


888651 i I 


72 


9i3oi4 


4-29 


086986 


42 


»9 


80181Q 
801973 


2.57 


888548 ! I 


72 


913271 


4-29 


086729 


41 


20 


2.57 


888444 1 I 


73 


913529 


4-29 


086471 


40 


21 


9-802128 


2.57 


9-888341 i I 


73 


9-913787 


4-29 


io-o862i3 


39 


22 


802282 


2.56 


888237 I 


73 


914044 


4-29 


085956 


38 


23 


802436 


2.56 


888134 1 


73 


914302 


4-29 


085698 


37 


24 


802 58q 
802743 


2-56 


888o3o I 


73 


914560 


4-29 


085440 


36 


25 


2-56 


887926 I 


73 


914817 


4-29 


o85i83 


35 


26 


802897 


2.56 


887822 I 


73 


915075 


4-29 


084925 


34 


27 


8o3o5o 


2-56 


887718 I 


73 


915332 


4-29 


084668 


33 


28 


803204 


2.56 


887614 I 


73 


915590 


4-29 


084410 


32 


0^ 


803357 


2.55 


887510 I 


73 


915847 


4-29 


084153 


3i 


So 


8o35ii 


2.55 


887406 I 


74 


916104 


4-29 


083896 


So 


3i 


9 -803664 


2.55 


9.887302 I 


74 


9-916362 


4-29 


10-083638 


ll 


32 


803817 


2.55 


887198 I. 


74 


916619 


4-29 


o8338i 


33 


803970 


2.55 


887093 I 


74 


916877 


4-29 


o83i23 


27 


34 


804123 


2.55 


886989 ; I- 


74 


917134 


4-29 


082866 


26 


35 


804276 


2.54 


886885 ! I. 


74 


917891 


4.29 


082609 


25 


36 


804428 


2-54 


886780 1 1. 


74 


917648 


4.29 


082352 


24 


ll 


804581 


2.54 


886676 1 1- 


74 


917905 


4-29 


082095 


23 


804734 


2.54 


886571 I. 


74 


918163 


4.28 


081837 


22 


h 


804886 


2.54 


886466 ! I- 


74 


918420 


4-28 


o8i58o 


21 


4o 


8o5o39 


2.54 


886362 I . 


75 


918677 


4-28 


o8i323 


20 


4i 


g-805191 


2-54 


9.886257 I. 


75 


9-918934 


4-28 


10-081066 


19 
18 


42 


805343 


2.53 


886i52 i 1- 


75 


919191 


4.28 


080809 


43 


805495 


2-5S 


886047 I • 


75 


919448 


4.28 


o8o552 


n 


44 


805647 


2.53 


885942 I . 


75 


919705 


4-28 


080295 


16 


45 


803799 


2.53 


885837 I . 


75 


919962 


4-28 


o8oo38 


i5 


46 


805951 


2.53 


885732 I. 


75 


920219 


4-28 


079781 


U 


tl 


806 io3 


2.53 


885627 I. 


75 


920476 


4-28 


079524 


i3 


806254 


2.53 


885522 I- 


75 


920733 


4-28 


079267 


12 


49 


806406 


2-52 


885416 1. 


75 


920990 


4-28 


079010 


II 


5c 


806557 


2-52 


885311 1. 


76 


921247 


4-28 


078753 


10 


5: 


9 806709 


2-52 


9-885205 I. 


76 , 


9.921503 


4.28 


10-078497 


t 


52 


806860 


2.52 


885100 1. 


76 


921760 


4-28 


078240 


53 


80701 I 


2-52 


884994 1 • 


76 


922017 


4-28 


077983 


1 


54 


807163 


2-52 


884889 I . 
884783 I . 


76 


922274 


4-28 ! 


077726 


6 


55 


807314 


2.52 


76 


922530 


4-28 


077470 


5 


56 


807465 


2.5l 


884677 1 . 


76 


922787 


4-28 


077213 


4 


u 


807615 


2-51 


884572 1 - 


76 


923044 


4-28 


076956 


3 


807766 


2.5l 


884466 I . 


76 


923300 


4-28 


076700 


2 


^ 


te? 


2.5l 


884360 1 . 


76 


923557 


4-27 


076443 


I 


60 


2.5l 


884254 I . 


77 


9238i3 


4-27 


076187 







Coeino 


D. 


Sine r 


>. ' Cotaiig. 


D. i 


Tang. 


IL 








(60 DB 


OKI 


1KB.) 









58 


(40 


DEGREES.) A 


TABLE 


] OF LOGARITHMIC 




~m' 


i Sine 


D. 


j Cosuio 


1 D. 


j Tting. 


D. 

1 


Cotang. 


60 





9-808067 


2-5l 


i 9.884254 


1-77 


9-928813 


i 4-27 


10.076187 


I 


j 808318 


i 3-5l 


' 884148 


1-77 


1 924070 


, 4-27 


073980 


5q 


3 


8o8368 


' 3-5l 


' 884042 


1-77 


924827 


I 4-27 


075673 ! 5$ 


3 


8o85i9 


2 -Do 


883936 


1-77 


1 924583 


1 4-27 


i 075417 57 ; 

! 075160 56 ' 


4 


808669 


2 00 


883829 


1-77 


1 924840 


4-27 


5 


808819 


2 -50 


883723 


1-77 


925056 


4-27 


j 074904 55 , 


6 


! 808969 


2 -50 


883617 


' 1-77 


923832 


4-27 


074648 54 i 


I 


809119 


2 -So 


883510 


1-77 


025609 


4-27 


074891 1 53 
074135 52 


809269 


2 • 5o 


883404 


1.77 


925863 


1 4-27 


9 


809419 


2-49 


i 883297 


1.78 


926122 


' 4-27 


078878 


5i 


IC 


809569 


3-49 


j 883191 


1.78 


926878 


; 4.27 


078622 


5o 


II 


9'8o97i8 


i 2-49 


9 -883084 


' '-78 


9-926684 


4-27 


10-073866 


49 


13 


809868 


: 2.49 


882077 


; 1-78 


926890 


4-27 


078110 


48 


i3 


810017 


: 2.49 


882871 


1.78 


927U7 


4-27 


072853 


47 


U 


810167 


2.4Q 


882764 


1.78 


927408 


4-27 


072597 


46 i 


i5 


8io3i6 


2.48 


882657 


1-78 


927659 


! 4-27 


072841 


45 : 


i6 


810465 


2-48 


88255o 


1.78 


927915 


4-27 


072085 


44 1 


\l 


810614 


3-48 


882443 


' 1.78 


928171 


4-27 


071829 
071573 


43 ! 


810763 


2-48 


882336 


1.79 


928427 


4-27 


42 


»9 


810913 


2-48 


882229 


1-79 


928688 


4-27 


071817 


41 


30 


811061 


2-48 


882121 


1-79 


928940 


4-27 


071060 


40 


31 


9-811210 


2-48 


9-882014 


1-79 


9.929196 


4.27 


10-070804 


39 
38 


33 


8ii358 


2-47 


881907 


1-79 


929432 


4-27 


070548 


33 


8ii5o7 


2-47 


88 1 799 


1-79 


929708 


4-27 


070292 
0700J6 


37 


24 


8ii655 


2-47 


881692 


1.79 


929964 


4-26 


36 


35 


811 804 


2-47 


88 1 584 


1-79 


980220 


4-26 


069780 


35 ! 


36 


811952 


2-47 


881477 


1.79 


980473 


4-26 


069525 


34 


3^ 


812100 


2-47 


881369 


1.79 


980781 


4-26 


069260 


33 


812248 


2-47 


881261 


1.80 


980987 


4-26 


0600 li 


32 


29 


812896 


2.46 


881 i53 


1.80 


981248 


4-26 


068757 


3i 


3o 


812044 


2-46 


881046 


1-80 


981499 


4-26 


o685oi 


3o 


3i 


9-8i2692 


2-46 


9-880938 


1-80 


9.981755 


4-26 


10-068245 


20 i 

28 


33 


812840 


3-46 


88o83o 


1-80 


982010 


4-26 


067990 
067784 


33 


812988 


2-46 


880722 


1.80 


982266 


4-26 


27 : 


34 


8i3i35 


2-46 


880613 


I -80 


982522 


4-26 


067478 


26 : 


35 


813283 


2-46 


88o5o5 


1-80 i 


982778 


4-26 


067222 


25 


36 


8i343o 


2-45 


880397 


I -80 ' 


988033 


4-26 


066967 


24 


ll 


813578 


3-45 


880289 


i-8i 


988289 


4.26 


066711 


23 


813725 


2-45 


880180 


i-8i 1 


988543 


4-26 


066455 


23 


39 


813872 


3.45 


880072 


1.81 


988800 


4-26 


066200 


31 i 


40 


814019 


2-45 


879963 


1-81 ' 


984056 


4-26 


065944 


30 1 


4^ 1 


9-814166 


2-45 


9.879855 


i.8i 


9.984811 


4-26 


10-065689 
965433 


18 i 


42 


8i43i3 


2-45 


879746 


1. 81 i 


984567 


4-26 


43 1 


81 1460 


2-44 


879637 


i.8i 


984823 


4-26 i 


o65i77 


17 ' 


44 


814607 


2-44 ' 


879529 


1.81 


983078 


4-26 ' 


064922 


16 


45 


814753 


2-44 i 


879420 . 


1. 81 


985838 


4-26 


064667 


i5 i 


46 


814900 


2-44 ! 


879811 


1. 81 j 


935589 


4-26 1 


06441 1 


14 i 


47 


81 5046 


2-44 


8^9202 


1.82 ' 


985844 


4-26 : 


0641 56 


i3 


48 


815193 


2-44 


870093 
878084 i 


1.82 


986100 


4-26 


068900 


12 


49 


815339 


2-44 


1-82 


986855 


4-26 


068645 


1 


5o 


81 5485 


2.43 


878875 


1.82 


986610 


4-26 


068890 





5i 


o-8i563i 


2-43 ; 


9-878766 i 


1.82 


9.986866 


4-23 


10 068184 j 


? 


53 


815778 i 


2-43 1 


878656 


1.82 


987121 


4-23 


062879 


53 


815924 


3-43 \ 


87S547 


1-82 ! 


987876 


4-23 


062624 


7 ! 


54 


816069 : 


3-43 : 


878488 


1.82 


937682 


4-25 


062868 


6 ' 


55 ' 


816215 


2-43 i 


878828 1 


1.82 


087887 


4-25 


002Il3 


f ! 

" 1 


5e ; 


8i636i 1 


3-43 1 


878219 i 


1.83 


988142 


4-25 1 


06 1 858 


4 \ 


57 ' 


8i65o7 


2-42 ! 


878109 ' 


1-83 


988898 


4-23 


06160a 


3 


58 


gi6652 


2-42 


877999 ' 


1.83 i 


988638 


4-23 , 


061847 


2 i 


59 i 


816798 ■ 


2-42 


877890 


1.83 


988908 


4-25 


061092 : I j 
060837 1 : 

Tang. ! M. ! 


60 i 


816943 1 


2-42 


877780 [ 


1.83 


989168 

1 


4-23 




Cosino 


D. ■ 


Sine 


D. 1 


Cotang. 1 


D. 








(49 


DEQRE 


ES) 







SINES AND TANGENTS. (41 DEGREES.; 



59 



M. 


Sine 


D. 


Cosine 


D. 


Taug. 


D. 


Cotaiig. 







g. 816943 


2-42 


9.877780 


1-83 


9.939163 


4-25 


10-060837 


60 


1 


817088 


2 


•42 


877670 


1-83 


939418 


4 


•25 


o6o582 


59 


2 


817333 


2 


•42 


877560 


1.83 


939673 


4 


•25 


060827 


58 


3 


817379 


2 


•42 


8-, 745o 


1-83 


939928 


4 


.25 


060072 


57 


4 


817024 


2 


•41 


8', 7340 


1-83 


940183 


4 


•25 


059817 


56 


5 


817668 


2 


•41 


8-, 723c 


1.84 


940438 


4 


•25 


059362 


55 


i 6 


817813 


2 


•41 


877120 


1.84 


940694 


i 


•25 


059806 


54 


' I 


817958 


^ 


■41 


877010 


1.84 


940949 


4 


•25 


oSooSi 


53 


8i8ic2 


2 


•41 


876899 
876789 


1.84 


941204 


4 


•25 


058796 


52 


9 


8itf247 


2 


41 


1.84 


941458 


4 


•25 


358542 


5i 


lO 


818392 


2 


41 


876678 


1.84 


941714 


4 


•25 


058286 


5o 


II 


9.8i8536 


2 


40 


9.876568 


1-84 


9.941968 


4 


•25 


10 .058032 


49 


la 


818681 


3 


40 


876457 


1.84 


942223 


4 


25 


057777 


48 


i3 


818825 


2 


40 


876347 


1-84 


942478 


4 


25 


057522 


47 


i4 


818960 
81911J 


2 


40 


876236 


1.85 


942733 


4 


25 


057267 


46 


i5 


2 


40 


876125 


1.85 


942988 


4 


25 


057012 


45 


i6 


819257 


2 


40 


876014 


1.85 


943243 


4 


25 


056757 


44 


17 


819401 


2 


40 


875904 


1-85 


943498 


4 


25 


o565o2 


43 


i8 


819545 


2 


39 


875703 


1-85 


943752 


4 


25 


056248 


42 


19 


819689 


2 


39 


875682 


1-85 


944007 


4 


25 


055993 


41 


20 


819832 


3 


39 


875571 


1-85 


944262 


4 


25 


055788 


40 


at 


9.819976 


2 


39 


9.875459 


1-85 


9-944517 


4 


25 


10.055488 


39 


22 


820120 


2 


39 


875348 


1.85 


944771 


4 


24 


055229 


38 


23 


820263 


2 


39 


875237 


1.85 


945026 


4 


24 


054974 


37 


24 


820406 


2 


39 


875126 


1.86 


945281 


4 


24 


054719 


36 


25 


82o55o 


2 


38 


875014 


1.86 


945535 


4 


24 


054465 


35 


26 


820693 
820836 


2 


38 


874903 


1.86 


945790 


4 


24 


054210 


34 


11 


2 


38 


874791 


1.86 


946045 


4 


24 


058955 


33 


820979 


2 


38 


874680 


1.86 


946299 


4 


24 


053701 


32 


29 


821122 


2 


38 


874568 


1.86 


946554 


4 


24 


053446 


3i 


3o 


821265 


2- 


38 


874456 


1.86 


946808 


4 


24 


058192 


3o 


3i 


9.821407 


3 


38 


9.874344 


1.86 


9.947063 


4 


24 


10.052987 


29 


32 


82i55o 


2 


38 


874232 


1.87 


947318 


4 


24 


052682 


28 


33 


821693 


2 


37 


874121 


1.87 


947372 


4 


24 


052428 


27 


34 


821835 


2 


37 


874009 


i.b7 


947826 


4 


24 


052174 


26 


35 


821977 


2 


37 


873896 


1.87 


948081 


4- 


24 


051919 


25 


36 


822120 


2 


37 


873784 


1.87 


948336 


4 


24 


o5i664 


24 


37 


822262 


2« 


37 


873672 


1.87 


948390 


4- 


24 


o5i4io 


23 


38 


822404 


2 


37 


873560 


1.87 


948844 


4- 


24 


o5ii56 


22 


39 


822546 


2 


37 


873448 


1.87 


949099 


4- 


24 


oSogoi 


21 


40 


822688 


2 


36 


873335 


1.87 


949353 


4- 


24 


050647 


20 


41 


9.822830 


2 


36 


9.873223 


1.87 
1.88 


9.949607 


4- 


24 


10 .050398 


\l 


42 


822972 


2 


36 


873110 


949862- 


4- 


24 


o5oi88 


43 


823114 


2 


36 


872998 


1.88 


950116 


4- 


24 


049884 


17 


44 


823255 


2 


36 


872885 


1.88 


950370 


4- 


24 


049680 


16 


45 


823397 


2 


36 


872772 


1-88 


950625 


i- 


24 


049875 


i5 


46 


823539 


2 


36 


872639 


1.88 


950879 


4- 


24 


049121 


14 


% 


823680 


2 


35 


872547 


1.88 


951 i33 


4- 


24 


048867 


i3 


823821 


2 


35 


872434 


1.88 


95 1 388 


4- 


24 


048612 


12 


49 


823963 


2 


35 


872321 


1.88 


951642 


4- 


24 


048358 


II 


5o 


824104 


2 


35 


872208 


1.88 


951896 


4- 


24 


048104 


10 


5i 


9.824245 


2 


35 


9-872095 


1.89 


9.952150 


4- 


24 


10-047850 





55 


824386 


2 


35 


871981 


1.89 


952405 


4- 


24 


047595 





53 


824527 


2 


35 


871868 


1.89 


932650 


4- 


24 


047841 


7 


^•4 


824668 


2 


34 


871755 


1.89 


95291J 


4- 


24 


047087 


6 


5f 


824808 


2 


34 


871641 


1.89 


953167 


4- 


23 


046833 


5 


5t 


824949 


2 


34 


871528 


1.89 


953421 


4- 


23 


046579 


4 


'!2 


8250Q0 


2 


34 


871414 


1.89 


953675 


4- 


23 


046325 


3 


82523c 


2 


34 


871301 


1.85 


953929 
9541 8J 


4- 


23 


046071 


a 


^ 


825371 


2 


34 


871187 


1.89 


4- 


23 


045817 


I 


60 


825511 


2.34 


871073 


1.90 


954437 


4-23 


045563 





Cosiuo 


D. 


Sine 


D. 


Cota]\g. 


_ J 


^•_J 


Tang. 


M. 



(48 DEaRXXo.) 



80 



(42 DEGREES.; A TABLE OF LOQaEITHMIC 



M. 

o 


Sine 


D. 


Cosine i D. 


Tang. 


D. 


CotAng. 


1 


9-8255ii 


2-34 


9-871073 I 


.90 


9-954487 


! 4'28 


10 045300 


60 


I 

2 


82565i 
825751 


2 
2 


33 
33 


870960 ! I 

870846 : I 


.90 
.90 


954691 
954945 


i 4 
4 


•23 

-28 


045309 59 
o45o53 • 58 


3 


825931 


2 


33 


870782 . I 


.90 


955200 


4 


•23 


044800 


5? 1 


4 


S26071 


2 


33 


870618 I 


.90 


955454 


4 


-23 


044546 


S6 


5 


82621 1 


2 


33 


870504 I 


.90 


955707 


4 


•23 


044298 


55 


6 


826351 


2 


33 


870890 I 


-90 


955961 


4 


-23 


044089 


54 . 


I 


826491 


2 


33 


870276 I 


90 


956215 


4 


-23 


048783 


53 


826631 


2 


.33 


870161 I 


-90 


956469 
956728 


4 


-23 


048381 


52 


9 


826770 


2 


32 


870047 I 


91 


4 


23 


048277 
048028 


5i 


IC 


826910 


2 


32 


869933 j I 


.91 


956977 


4 


-23 


5o 


II 


9-827049 


2 


32 


9-869818 I I 


91 


9-957281 


4 


23 


10.042769 
0425 1 5 


49 
48 


12 


827189 
827828 


2 


32 


869704 I 


91 


957485 


4 


23 


i3 


2 


32 


869589 I 


-91 


937789 


4 


23 


042261 


47 


14 


827467 


2 


32 


869474 I 


91 


937998 


4 


23 


042007 


46 


i5 


827606 


2 


32 


869860 I 


91 


958246 


4 


23 


041754 


45 


i6 


827745 


2 


32 


869245 I 


91 


9585oo 


4 


23 


04 1 5oo 


44 


\l 


827884 


2 


3i 


869130 I 


.91 


958754 


4 


-23 


041246 


43 


828023 


2 


3i 


869015 1 I 
868900 I 


.92 


959008 


4 


-23 


040992 


42 


»9 


828162 


2 


3i 


92 


959262 


4 


23 


040788 


41 


20 


828301 


a 


3i 


868785 I 


92 


959516 


4 


-23 


040484 


40 


21 


9-828439 


2 


3i 


9-868670 1 I 


92 


9.959769 


4 


28 


10-040281 


39 


22 


828578 


2 


3i 


868555 1 


92 


960028 


4 


23 


089977 


38 


23 


828716 


2 


3i 


868440 j I 


92 


960277 


4 


23 


089728 


37 


24 


828855 


2 


3o 


868324 1 I 


92 


960581 


4 


28 


089469 


36 


25 


828993 
829131 


2 


3o 


868209 1 I 


92 


960784 


4 


28 


089216 


35 


26 


2 


3o 


868098 I 


92 


961088 


4 


23 


088962 


34 


11 


829269 


2 


3o 


867078 1 


93 


961291 


4 


23 


088709 


33 


829407 


2 


3o 


867862 I 


93 


961545 


4 


28 


088453 


32 


29 


829545 


2 


3o 


867747 ; I 


93 


961799 


4 


23 


088201 


3i 


3o 


829683 


2 


3o 


867681 I 


93 


962052 


4 


28 


087948 


3o 


3i 


9.829821 


2 


29 


9-867515 1 


93 


9-962806 


4 


23 


10-087694 


29 


32 


829959 


2 


29 


867899 i I 


93 


962560 


4 


23 


087440 


28 


33 


83oo97 


2 


29 


867288 , I 


93 


962818 


4 


23 


087187 


27 


34 


830234 


2 


29 


867167 i I 


93 


968067 


4 


23 


086933 


26 


35 


83o372 


2 


29 


867051 ' 1 


93 


968820 


4 


23 


086680 


25 


36 


83o5o9 


2 


29 


866935 i I 
866819 ! J 
866708 i I 


94 


963574 


4 


23 


086426 


24 


37 


830646 


2 


29 


94 


968827 


4 


28 


086178 


23 


38 


830784 


2 


29 


94 


964081 


4 


28 


085919 


22 


39 


830921 


2 


28 


866586 1 I 


94 


964885 


4 


23 


o35663 


21 


40 


83io58 


2 


28 


866470 1 I 


94 


964588 


4 


22 


o354ia 


20 


41 


9-83;i95 


2 


28 


9-866353 i 


94 


9 - 964842 


4 


22 


io-o35i58 


10 
18 


42 


83i332 


2 


28 


866287 I 


94 


965095 


4 


22 


o349o5 


43 


831469 


2 


28 


866120 I 


94 


965349 


4 


22 


084651 


\l 


44 


83 1 606 


2 


28 


866004 I 


95 


965602 


4 


22 


084898 


45 


831742 


2 


28 


865887 I 


95 


965855 


4 


22 


084145 


i5 


46 


831879 


2 


28 


865770 1 I 


95 


966105 


4 


22 


088891 


14 


S 


83201D 


2 


27 


865653 : I 


95 


966862 


4 


22 


033688 


i3 


832152 


2 


27 


865586 j I 


95 


966616 


4 


22 


088384 


13 , 


49 


832288 


2 


27 


865419 ! I 


95 


966869 


4 


22 


o83i3i 


II 


5o 


832425 


2 


27 


865302 I 


95 


967123 


4- 


22 


082877 


10 


5i 


0-832561 


2 


27 


9-865185 I 


95 


9-967876 


4- 


22 


10-082624 


I 


52 


882697 


2 


27 


865o68 1 I- 


95 


967629 


4- 


22 


082871 


53 


832833 


2 


27 


864950 i I- 


95 


967888 
968186 


4- 


22 


082117 


I 


54 


832969 
833103 


2 


26 


864833 1 I- 


96 


4- 


22 


o3i864 


55 


2 


26 


864716 I- 


96 


968889 
968648 


4- 


22 


081611 


5 


56 


833241 


2 


26 


864598 I- 


96 


4- 


22 


o3i357 


4 


ll 


833377 


2 


26 


864481 I- 


96 


068896 


4- 


22 


081104 


3 


833512 


2 


26 


864363 I - 


96 


969149 
969408 


4- 


22 


o8o85i 


3 


59 


833648 


2 


26 


864245 I - 


96 


4- 


22 


o3o597 


I 


60 


833783 


2-26 


864127 I- 


96 


969656 


4-22 


080844 





Coeine 


D. 


Sine E 


Cotang. 


D. 


Tang. 


^, 



(47 DEGREES.) 



SINES -IND TANGENTS. (43 DEGREES.) 



61 



M. 


Sine 


D. 


Cosine D. 


Tang. 


D. 


Cotar g. 







9 833783 


2-26 


9-864127 1 


-96 


9.969656 


4-22 


10-080844 


60 


I 


833919 


2-25 


864010 I 


96 


969909 


4 


-22 


080091 
029888 


u 


3 


8i^o54 


2-25 


868892 1 


97 


970162 


4 


22 


3 


834 I 8q 


2-25 


868774 1 


97 


^70416 


4 


22 


029584 


u 


4 


834325 


2-25 


868656 1 I 


97 


970669 


4 


22 


029881 


6 


834460 


2-25 


868588 '■ 1 


97 


970922 


4 


22 


029078 


55 


6 


834595 
834''lo 


2-25 


868419 1 


97 


97XX75 


4 


22 


028825 


54 


3 


2-25 


863301 1 


97 


971429 


4 


22 


02857X 


53 


834805 


2-25 


868x88 I 


97 


971682 


4 


22 


028818 


52 


9 


834999 
835i34 


2 24 


868064 I 


97 


971985 


4 


22 


028065 


5i 


10 


2-24 


862946 1 


98 


972188 


4 


22 


027812 


5o 


II 


9.835269 
8354o3 


2-24 


9.862827 X 


98 


9-972441 


4 


22 


10-027559 


49 


12 


2-24 


862709 I 


98 


972694 


4 


22 


027806 


48 


i3 


835538 


2-24 


862590 1 


9b 


97294^. 


4 


22 


027052 


47 


14 


835672 


2-24 


862471 I 


98 


978201 


4 


22 


026799 


46 


i5 


835807 


2-24 


862853 I 


98 


973454 


4 


22 


026546 


45 


i6 


835941 


2-24 


862284 i I 


98 


978707 


4 


22 


020298 


44 


17 


886075 


2-28 


862xx5 1 


98 


978960 


4 


22 


026040 


43 


i8 


836209 


2-23 


861996 1 1 


98 


974218 


4 


22 


025787 


42 


19 


836343 


2-28 


861877 ' I 


98 


974466 


4 


22 


025584 


41 


20 


836477 


2-28 


861758 X 


99 


974719 


4 


22 


025281 


40 


21 


9-83661 1 


2-23 


9-861688 i X 


99 


9-974973 


4 


22 


I0-025027 


39 


22 


836745 


2-23 


86x5:9 i x 


99 


975226 


4 


22 


024-;74 


39 


23 


836878 


2-23 


861400 I 


99 


975479 


4 


22 


024521 


37 


24 


837012 


2-22 


861280 I 


99 


975782 


4 


22 


024268 


86 


25 


837146 


2-22 


86x161 I 


99 


975985 


4 


22 


0240x5 


35 


26 


837279 


2-22 


861041 1 


99 


976288 


4 


22 


028762 


34 


11 


837412 


2-22 


860922 I 


99 


976491 


4 


22 


>285o9 


33 


S37546 


2-22 


860802 1 


99 


976744 


4 


22 


023256 


32 


?9 


S37679 


2-22 


860682 2 


00 


976997 


4 


22 


028008 


3i 


So 


837812 


2-22 


86o562 2 


00 


97725o 


4 


22 


022750 


3o 


3i 


9-837945 


2.22 


9.860442 2 


00 


9.977508 


4 


22 


10-022497 


29 


32 


838078 


2.21 


860822 2 


00 


977756 


4 


22 


022244 


28 


33 


838211 


2-21 


860202 2 


00 


978009 


4 


22 


02199X 


27 


34 


838344 


2-21 


860082 2 


00 


978262 


4 


22 


021788 


26 


35 


838477 


2-21 


859062 2 


00 


9785x5 


4 


22 


02x485 


25 


36 


8386x0 


2-21 


859842 2 


00 


978768 


4 


22 


021282 


24 


37 
38 


838742 


2-21 


859721 2 


01 


979021 


4 


22 


020979 


23 


888875 


2-21 


859601 2 


01 


979274 


4 


22 


020726 


23 


39 


889007 


2-21 


859480 2 


01 


979527 


4 


22 


020478 


21 


40 


889140 


2-20 


859860 2 


01 


979780 


4 


22 


020220 


20 


41 


9-889272 


2-20 


9.859289 2 


ox 


9.980083 


4 


22 


10-0x9967 


\t 


42 


889404 


2-20 


859119 2 


ox 


980286 


4 


22 


019714 


43 


889586 


2-20 


858998 2 


01 


980538 


4 


22 


019462 


n 


44 


889668 


2-20 


858877 2 


01 


980791 


4 


2X 


019209 


16 


45 


889800 


2-20 


858756 2 


02 


981044 


4 


21 


0x8956 


x5 


46 


889982 


2-20 


858635 2 


02 


981297 


4 


21 


0x8708 


14 


% 


840064 


2-19 


8585x4 2 


02 


98x550 


4 


2X 


018450 


i3 


840196 


2-19 


858898 2 


02 


981808 


4 


21 


0x8197 


12 


49 


840828 


2-19 


858272 2 


02 


982056 


4 


21 


0x7944 


11 


5o 


840459 


2-19 


858i5i 2 


02 


982809 


4 


21 


017691 


10 


5i 


9-840591 


2-19 


9-858029 2 


02 


9-982562 


4 


21 


10017488 


9 


52 


840722 


2-19 


857908 2 


02 


982814 


4 


21 


0x7x86 


§ 


53 


840854 


2.19 


857786 2 


02 


988067 


4 


2X 


016988 


I 


54 


840985 


2.10 
2.16 


857665 2 


o3 


988820 


4 


21 


0x6680 


55 


841116 


857543 2 


o3 


988578 


4 


23 


0x6427 


5 


5t 


841247 
841878 


2-18 


857422 2 


o3 


988826 


4 


21 


0x6174 


4 


u 


2-18 


857800 2 


o3 


984079 


4< 


21 


015921 


3 


84: 509 


2.18 


857178 2 
857056 2 


o3 


98488 X 


4 


21 


015669 


a 


59 


841640 


2.18 


o3 


984584 


4 


21 


0i54i6 


I 


60 


841771 


2.18 


856984 2 


o3 


984887 


4-21 


oi5i63 





Cosine 


D. 


Sine r 


r 


Cotanor. 


D. 


Tang. 


M. 



(46 DBORESS.) 



S2 



(44 DEGREES.) A TABLE OF LOGARITHSJIC 



M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 
io-oi5i63 


1 


o 


9-841771 


2.18 


9.856934 


2-o3 


9-984837 


4-21 


60 


I 


841902 


3.18 


856812 


2 


o3 


985090 


4 


31 


014910 


U 


3 


842033 


2 


18 


856690 


2 


04 


985343 


4 


21 


014657 


3 


842163 


2 


17 


856568 


2 


04 


9855^6 


4 


21 


014404 


u 


4 


842294 


2 


17 


856446 


2 


04 


985848 


4 


21 


oi4i52 


5 


842424 


2 


17 


856323 


2 


04 


986101 


4 


21 


013899 


55 


6 


842555 


2 


17 


856201 


2 


04 


986354 


4 


21 


013646 


54 


I 


842685 


2 


17 


856078 


2 


04 


986607 


4 


21 


013393 


53 


842815 


2 


17 


855956 


2 


04 


986860 


4 


21 


oi3i4o 


52 


9 


842946 


2 


17 


855833 


2 


04 


987112 


4 


21 


012888 


5i 


10 


843076 


2 


n 


8557 1 1 


2 


o5 


987365 


4 


21 


012635 


5o 


II 


9.843206 


2 


16 


9-855588 


2 


o5 


9-987618 


4 


21 


10-012383 


45 


13 


843336 


2 


16 


855465 


2 


o5 


987871 


4 


21 


012129 


48 


i3 


843466 


2 


16 


855342 


2 


o5 


988123 


4 


21 


311877 


47 


14 


843595 


2 


16 


855219 


2 


o5 


988376 


4 


21 


011624 


46 


i5 


843725 


2 


16 


855096 


2 


o5 


9S8629 


4 


21 


011371 


45 


i6 


843855 


2 


.16 


854Q73 


2 


o5 


988882 


4 


21 


011118 


44 


\l 


843984 


2 


16 


854850 


2 


o5 


989134 


4 


21 


010866 


43 


844114 


2 


i5 


854727 


2 


c6 


989387 


4 


21 


oio6i3 


42 


19 


844243 


2 


i5 


854''-'-3 


2 


06 


989640 


4 


21 


oio36o 


41 


30 


844372 


2 


i5 


854480 


2 


06 


989893 


4 


21 


010107 


40 


31 


9 -844502 


2 


i5 


9-854356 


2 


06 


9.990145 


4 


21 


10-009855 


39 


33 


844631 


2 


i5 


854233 


2 


06 


990398 


4 


21 


009602 


38 


23 


844760 


2 


i5 


854109 


2 


06 


990601 


4 


21 


009349 


37 


24 


844889 


2 


i5 


853986 


2 


06 


990903 


4 


21 


009097 


36 


25 


845018 


3 


i5 


853862 


2 


06 


99ii56 


4 


21 


008844 


35 


36 


845147 


3 


i5 


853738 


2 


06 


991409 


4 


21 


008591 


34 


11 


845276 


3 


14 


853614 


2 


07 


991662 


4 


21 


008338 


33 


845405 


3 


14 


853490 


2 


07 


991914 


4 


21 


008086 


?2 


29 


845533 


2 


14 


853366 


2 


07 


992167 


4 


21 


007833 


3i 


3o 


845662 


2 


14 


853242 


2 


07 


992420 


4 


21 


007580 


3o 


3i 


9-845790 


2 


14 


9-853ii8 


2 


07 


9-992672 


4 


21 


10-007328 


29 


32 


845919 


2 


14 


802994 


2 


07 


992925 


4 


21 


007075 


38 


33 


846047 


2 


14 


852869 


2 


07 


993178 


4 


21 


006822 


27 i 


34 


846175 


2 


14 


852743 


2 


07 


993430 


4 


21 


006570 


36 


35 


846304 


2 


14 


852620 


2 


07 


993683 


4 


21 


oo63i7 


25 


36 


846432 


2 


i3 


852496 


2 


08 


993936 


4 


21 


006064 


2»^ 


ll 


846560 


2 


i3 


852371 


2 


08 


994189 


4 


21 


Do58 1 1 


23 


846688 


2 


i3 


852247 


2 


08 


994441 


4 


21 


X)5559 


22 


39 


846816 


2 


i3 


852122 


2 


08 


994694 


4 


21 


oo53o6 


21 i 


4o 


846944 


2 


i3 


851997 


2 


08 


994947 


4 


21 


oo5o53 


20 1 


41 


9-847071 


2 


i3 


9.851872 


2 


08 


9.995199 


4 


21 


10-004801 


1 
19 
10 . 


43 


847199 


2 


i3 


851747 


2 


08 


995402 


4 


21 


004548 


43 


847327 


2 


i3 


85x622 


2 


08 


995705 


4 


21 


004295 


n i 


44 


847454 


2 


12 


851497 


2 


09 


995957 


4 


21 


004043 


16 , 


45 


847582 


2 


12 


85i372 


2 


09 


996210 


4 


21 


003790 


i5 : 


46 


847709 


2 


12 


851246 


2 


09 


996463 


4 


21 


003537 


14 


47 


847836 


2 


13 


85ii2i 


2 


09 


996715 


4 


21 


003285 


i3 j 


48 


847964 


2 


12 


800996 


2 


09 


996968 


4 


31 


oo3o3: 


13 ; 


49 


848091 


2 


12 


850870 


2 


09 


997221 


4- 


21 


002779 


'1 : 


5o 


848218 


3 


12 


850745 


2 


09 


997473 


4- 


21 


002527 


ro ; 


5i 


9-848345 


3 


12 


9-850619 
85o493 


2 


09 


9.997726 


4- 


21 


IO-002;274 


? 


53 


848472 


2 


II 


2 


10 


997979 


4- 


21 


002021 


53 


848599 


2 


II 


85o368 


2 


10 


998231 


4 


21 


001760 


I 


54 


848726 


2 


" 


850242 


2 


10 


998484 


4- 


21 


ooi5i6 


55 


84S852 


3 


II 


85oii6 


2 


IC 


998737 


4 


21 


001263 


5 


56 

ll 


848979 
849106 


3 
2 


II 
II 


849990 
849064 


2 
2 


10 
10 


998989 
999242 


4- 
4 


21 
21 


OOIOIl 

ooopS 
ooooo5 


4 
3 


849233 


3 


II 


849738 


3 


10 


999490 


4 


21 


2 


59 


849359 


a 


II 


8496 1 1 


2 


10 


999748 


4 


21 


00O253 


1 


60 


849485 


3-II 


849485 


2-10 


10- 000000 


4-31 


10- 000000 


u 1 




Coeine 


D. 


Sine 


D. 


Cotang. 


D. 


T^ftMr. 


JLj 



(45 DEGREB8.) 



A TABLE OF NATURAL SINES. 



98 



M 




Deg. 


1 Deg. 


2 Deg. 


8 Deg. 


4 Deg. 


M 


S. 

00000 


c. s. 


S. 


as. 


S. 


as. 


S. 


as. 


s. 


as. 


Uuit. 


01745 


99985 


03490 


99980 


o5234 


99868 


06976 


99756 


60 


I 


00029 


I -0000 


01774 


99984 


o35i9 


99988 


o5263 


99861 


07005 


99754 59 


3 


ooo58 


I -0000 


oi8o3 


99984 


03548 


99987 


00292 


99860 


07084 


99753; 58; 


3 


00087 


1 -0000 


oi832 


99983 


08077 


99986 


o532i 


99858 


07068 


99750 


57 i 


4 


001 16 


I -0000 j 01862 


99983 


08606 


99935 


o535o 


99857 


07092 


99748 


56' 


5 


00145 


1. 0000 j 01891 


99982 


o8635 


99984 


o5879 


99855 


07121 


99746 


551 


6 


00175 


I • 0000 ! 01920 


99982 


08664 


99988 


o54o8 


99854 


07i5o 


99744 


54 j 


7 


00204 


i.ooool 


01940 


99981 


08698 


99982 


05487 


99802 


07179 


99742 


53 


8 


00233 


I .0000 


01978 


99980 


08728 


99981 


05466 


99801 


07208 


99740 


52 i 


9 


00262 


1.0000 


02007 


99980 


08752 


99980 


05495 


99849 


07287 


99788 


5i 


10 


00291 


I. 0000 


02o36 


99979 


08781 


99929 


o5524 


99847 


07266 


99786 


5o 


II 


00320 


99999 


o2o65 


99979 


08810 


99927 


05553 


99846 


07295 


99734 


49 


13 


oo34o 


99999 


02094 


99978 


08889 


99926 


05582 


99844 


07824 


99781 


48 


i3 


00378 


99999 


02123 


99977 


08868 


99925 


00611 


99842 


07353 


99729 


47 


14 


00407 


99999 


02l52 


99977 


08897 


99924 


o564o 


99841 


07882 


99727 


46; 


i5 


00436 


99999 


02181 


99976 


08926 


99923 


o566q 


99889 


07411 


99725 


45 


i6 


00465 


99999 


02211 


99976 


08955 


99922 


05698 


99888 


07440 


99723 


44i 


\l 


00493 


99999 


02240 


99970 


08984 


99921 


05727 


99886 


07469 


99721 


43 i 


oo524 


99999 


02269 


99974 


04018 


99919 


05756 


99884 


07498 


99719 


42 1 


«9 


oo553 


99998 


02298 


99974 


04042 


99918 


05785 


99833 


07527 


99716 


41' 


30 


00582 


99998 


02327 
02356 


99973 


04071 


99917 


o58i4 


99881 


07556 


99714 


40. 


31 


006 1 1 


99998 


99972 


04100 


99916 


o5844 


99829 


07585 


99712 


39! 


33 


00640 


99998 


02385 


99972 


04129 


99915 


05878 


99827 


07614 


99710 


38 


33 


00660 


99998 


02414 


99971 


04159 


99913 


C5902 


99826 


07643 


99708 


37 


34 


00698 


99998 


02443 


99970 


04188 


99912 


05981 


99824 


07672 


99705 


36! 


35 


00727 


99997 


02472 


99969 


04217 


99911 


05960 


99822 


07701 


99703 


351 


36 


00756 


99997 


025oi 


99960 


04346 


99910 


05980 


99821 


07780 


99701 


341 


11 


00785 


99997 


o253o 


99968 


04275 


99909 


06018 


99819 


07759 


99699 


331 


00814 


99997 


02560 


99967 


o43o4 


99907 


06047 


99817 
99810 


07188 
07817 


99696 


32' 


39 


00844 


99996 


02589 


99966 


04333 


99906 


06076 


99694 


3ij 


3o 


00873 


99996 


02618 


99966 


04862 


99905 


o6io5 


99818 


07846 


99692 


30 j 


3i 


00902 


99996 


02647 


99965 


04391 


99904 


06184 


99812 


07875 


99689 


29! 

?8' 


33 


00931 


99996 


02676 


99964 


04420 


99902 


06168 


99810 


07904 


l& 


33 


00960 


99995 


02705 


99963 


o444q 


99901 


06192 


99808 


07988 


271 


34 


00989 


99995 


02734 


99963 


04478 


99900 


06221 


99806 


07962 


99683 


26 1 


35 


01018 


99990 


02763 


99962 


04507 


99898 


o635o 


99804 


07991 


99680 


25, 


36 


01047 


99990 


02792 


99961 


04536 


99897 


06279 


99808 


08020 


99678 


24 


37 


01076 


99994 


02821 


99960 


04565 


99896 


06800 


9,9801 


08049 


99676 


23 


38 


oiio5 


99994 


02850 


99959 


04594 


99894 


06887 


99799 


08078 


99678 


23 


39 


01134 


99994 


02879 


99959 


04628 


99898 


06866 


99797 


08107 


99671 


21 


40 


01164 


99993 


02908 


99958 


04653 


99892 


06395 


95795 


08186 


99668 


20 


41 


01193 


99993 


02938 


99957 


04682 


99890 


06424 


99793 


081 65 


99666 


19 


43 


01222 


99993 


02967 


99956 


047 1 1 


Q9880 


06453 


99792 


08194 


99664 


18 


43 


0125l 


99992 


02996 


99955 


04740 


99888 


06482 


99790 


08228 


99661 


17 


44 


01280 


99992 


o3o25 


99904 


04760 


99886 


o65ii 


99788 


08252 


99659 


16 


45 


01 309 99991 


o3o54 


99953 


04798 


99885 


06540 


99786 


08281 


99657 


i5 


46 


oi338 99991 


o3o83 


99952 


04827 


99883 


o65oo 


99784 


o83io 


99654 


14 


47 


01367 9999' 


03ll2 


99952 


04856 


99882 


06598 


99782 


08889 


99652 


i3 


48 


01396 99990 


o3i4i 


99951 


04885 


99881 


06627 


99780 


08868 


99649 


12 


49 


01425 999QO 


08170 


9q95o 


04914 


99879 


06656 


99778 


08897 


99647 


11 


5o 


01404 99989 


0319Q 


99940 


04948 


99878 


06685 


99776 


08426 


99644 


10 


fi 


01483 99989 


03228 


99948 


04972 


99876 


06714 


99774 


08455 


99642 


0. 


53 


oi5i3 99989 


03257 


99947 


o5ooi 


99875 


06743 


99772 


08484 


99689 


81 


53 


01542! 99988 


03286 


99946 


o5o3o 


99878 


06778 
06802 


99770 


o85i3 


99687 


7 


54 


01571 99988 


o33i6 


99945 


oSoSg 


99872 


99768 


08542 


99685 


6 


55 


01600 99987 


03345 


99944 


o5o88 


99870 


06881 


99766 


08571 


99683 


5 


56 


01620 99987 


03374 


99943 


o5ii7 


99869 


06860 


99764 


08600 


99680 


4 


u 


oi658 99986 


o34o3 


99942 


o5i46 


99867 


06889 


99762 


08629 


99627 

99625 


3 


01687 j 99986 


03433 


99941 


o5i75 


99866 


06918 


99760 


08658 


a 


59 
M 


017161 99985 


03461 

C. S. 


99940 


o52o5 


99864 


06947 

as. 


99758 


08687 

as. 


99623 

s 


I 


c. s. i s. 


S. 


as. 


s. 


s. 


89 Deff. 


88 J 


Deg. 


87 Deg. 


86jDeg^__ 


85 Deg. 



54 



A TABLE OF NATURAL SINES- 



M 


6 Deg. 


6 Deg. 


1 V Deg. 


8 Deg. 


9 Deg. 


r] 


S. 1 c. s. 


S. 


C. S. 


! S. ; C. S. 


S. 


C S 


s. 1 c s. 


M| 





08716 99619 


j 10453 


99452 


12187 99255JJ 13917 


99027 


1 5643 i 98769! 60 1 


I 


08745 


99617 


1 10482 


99449 


12216 99231 i| 13946 


99023 


1 5672 i 98764 


i? 


3 


08774 


99614 


io5ii 


99446 


1 12245 99248; 13975 


99019 


15701 1 98760 


3 


08803 


99612 


io54o 


99443 


12274 99244' 14004 


99015 


i573o! 98755 


57 


4 


o883i 


99609 


10569 


99440 


i23o2, 99240J i4o33 


99011 


i5758| 98751 


56 


5 


08860 


99607 


10597 


99437 


. i233i: 99237; 14061 


99006 


15787 98746 


55 


6 


08880 


99604 


10626 


99434 


' 12360 992331 14090 


99002 


i58i6! 98741 


54 


I 


08918 


99602 


10655 


9943 1 


, 12389 9923o'; 14119 


98998 


15845; 98737 


53 


08947 


99399 


; 10684 


99428 


1 12418 99226 j 14148 


98994 


15873 98732 


52 


9 


08976 


99396 


1 07 1 3 


99424 


j 12447 


99222'! 14177 


98990 


15902! 98728 


5i| 


10 


09005, 99394 


i 10742 


99421 


j 12476 


99219J i42o5 


98986 


15931 98723 


5oi 


II 


09034 99591 


10771 


99418 


j i25o4 


99213;. 14234 


.)8982 


15959 98718 


49 


12 


09063 j 99388 


10800 


99415 


1 12533 


99211; 14263 


98978 


15988J 98714 


48 


i3 


09092 99386 


10829 


99412 


1 12362 


99208! 14292 


98973 


16017 98709 


47 


14 


091 21 99583 


10858 


99409 


12591 


99204I 14320 


98969 


16046 98704 


46 


i5 


09150 99580 


10887 


99406 


12620 


99200! 14349 


98963 


16074 98700 


45 


i6 


09179I 99578 


10916 


99402 


1 12640 


99197I! 14378 


98961 


i6io3 98695 


44 


\l 


092081 99375 


10945 


99399 


1 12678 


99io3;i 14407 


98937 


16132} 98690 


43 


09237 99572 


10973 


99396 


1 12706 


99189! 14436 


98953 


16160 98686 


42 


19 


092661 99570 


11002 


99393 


12735 


99186I T4464 


98948 


16189I 9S681 


41 


20 


092951 99367 


iio3i 


99300 


12764 


991821 14493 


98944 


16218 98676 


40 


21 


09324, 99364: 


11060 


993S6 


12793 


99178I 14522 


98940 


16246 98671 


39 


22 


09353 1 99362 j 


11089 


99383 


12822 


99173! i455i 


98936 


i62t5| 98667 


38 


23 


09382 99339 


11118 


99380 


i285i 


99171 14580 


98931 


i63o4 


98662 


37 


24 


0941 1 1 99536 


11 147 


99377 


12880 


99167 14608 


98927 


16333 


98657 


36 


25 


09440 99553 


11176 


99374 


1 2908 


99163 14637 


98923 


i636i 


98652 


35 


26 


09469 99551 


11205 


99370 


12937 


99160 14666 


98919 


16390 


98648 


34 


11 


094981 99548 


11234 


99367 


12966 


991 56 14695 


98914 


16419 


98643 


33 


09527 99345 
09556 99542 


11263 


99364 


12995 


99152 


14723 


98910 


16447 


98638 


32 


29 


11291 


99360 


i3o24 


99148 


14752 


98906 


16476 


98633 


3i 


3o 


09383 


99540 


ll320 


99357 


i3o53 


99144 


14781 


98902 


i65o5 


98629 


3o 


3i 


09614 


995371 


11349 


99354 


i3o8i 


99141 


14810 


98897 


16533 98624 


29 


32 


09642 


99534 
99531 
99528 


11378 


99351 


i3iio 


99137 


14838 


98893 


16562 98619 


28 


33 


09671 


11407 


99347 


i3i39 


99133 


14867 


98889 


16591 98614 


27 


34 


09700 


11436 


99344 


i3i68 


99129: 14896 


98884 


16620 98609 


26 


35 


09720 99526 


11465 


99341 


i3i97 


99125 


14923 


98880 


16648 98604 


25 


36 


09758 99323 


11494 


99337I 


13226 


99122 


14954 


98876 


16677 98600 


24 


ll 


09787 99520 


ii523 


99334' 


13234 


99118 


14982 


98871 


16706 98595 


23 


09816 99517 


ii552 


99331 i 


13283 


991 14 


i5oii 


98867 


16734 98590 


32 


39 


09845 99514 


ii58o 


993271 


i33i2 


99110 


i5o4o 


98863 


16763 98585 


21 


40 


09874 9951 1 j 


11609 


99324 


13341 


99106 


15069 


98858 


16792 98580 


20 


41 


09903 99508 


ii638 


99320 


13370 


99102 


i5o97 


98834 


168201 98575 


19 


42 


0993 2 1 993061 


11667 


99317 


13399 


99098 


i5i26 


98849 


168491 98570 


10 


43 


0906 1 1 995o3l 


1 1696 


09314 


13427 


99094 


I3i55 


98845 


16878] 98565 


\l 


44 09990' 903001 


1x723 


99310 


13456 


990QI 


i5i84 


98841 


16906! 98561 


45 


10019 994971 


11754 


99307 


13485 


99087 1 13212 


98836 


16935 98556 


i5 


46 10048 99494 


11783 


993o3 


i35i4 


99083 1 5241 


98832 


16964 985511 14 


4"' 1 10077I 99491 


11812 


99300 


13543 


99079 15270 


988271 


16992 98546' i3 


i8 1 10106^ 99488! 


1 1 840 


99297 


13572 


99075 


15292 


98823- 


17021' 98541, 12 


49 


ioi33! 994851 


11869 


99293 


i36oo 


99071 


15327 


98818^ 


i7o5o: 985361 ir 1 


5o 


10164' 99482! 


11898, 


99290] 


13629 


99067 1 


15356 


98814, 


17078 98531 10' 


5i 


10192; 99479' 


11927' 


99286 


13658 


99063 j 


15385 98809' 


17107' 98326 


§ 


52 


1022 1 99476 


11956 


99283 1 


130S7 


99059: 13414 98805 1 


I7i36i 98521 


53 


io25o| 99473 


11985 


99279, 


13716 


99033 134421 98800J 


17164' 98516 


I 


54 


102791 99470 


12014 


99276 


13744 


9905 1 


134711 9879^1 


17193I 98311 


55 


io3o8i 99467' 


12043 


99272 


13773, 


99047 


i55oo 987911 


17222J 98506 


5 


56 


io337J 99464^ 


12071 


99269 


i38o2 


99043 i5529| 


98787: 


17250' 98501 


4 


ll 


io366i 99461 


12100 


99265 


i333i! 


99039! 15557I 


98782 


173 Q 98496 


3 


10395, 994581 


12129 


99262 


i386o 


99033 j 


1 5586 


9877SI 


17308 98491 
17336 98486 


2 


59 


104241 99455; 

c. s. i S. 1 


i2i58 

C. S. ! 


99258 

S. 


13889 


9903 1 1 


i56i5 


98773; 


I 
"M 


C. S. 1 


s. 1 


C. S. 


s- ! 


c. s. s. 


841 


)eK. 1 


88 Deg. 1 


82 Deg. 1 


81 Deg 1 


80 Deg 



A TABLE OF NATURAL SINES. 



66 



M 


10 Deg. 


11 Deg. 


12 Deg. 


18 Deg. 


14 Deg. 


u 


S. 


c. s. 


S. 


c. s. 


S. 
20791 


C. S. 
97815 


S. 
22495 


C. S. 


S. 


C. S. 





17365 


98481 


1 908 1 


98163 


97437 


24192 


97080 


601 


I 


17393 


98476 


19109 


98157 


20820 


97809 


22523 


9743o 


24220 


97023 


u 


a 


17422 


98471 


19138 


98152 


20848 


97803 


22552 


97424 


24249 


97015 


3 


1 745 1 


98466 


19167 


98146 


20877 


97797 


2258o 


97417 


24277 


97008 


57 


4 


17479 


9B46I 


19195 


98140 


20905 


97791 


22608 


97411 


243o5 


97001 


56 


5 


17508 


98455 


19224 


98135 


20933 


97784 


22637 


97404 


24333 


96994 


55 


6 


17537 


98450 


19252 


98129 


20962 


97778 


22665 


97898 


24862 


96987 


54 


I 


17565 


98445 


19281 


98124 


20990 


97772 


22693 


97891 


24890 


96980 


53, 


17594 


98440 


19309 


98II8 


21019 


97766 


22722 


97384 


24418 


96973 


52! 


9 


17623 


98435 


19338 


98112 


21047 


97760 1 22750 


97378 


24446 


96966 


5i 


10 


17651 


98430 


19366 


98107 


21076 


97754 


22778 


97871 


24474 


96959 


5o 


II 


17680 


98425 


19395 


98101 


2 II 04 


97748 


22807 


97865 


245o3 


96952 


g: 


12 


17708 


98420 


19423 


98096 


2ll32 


97742 


22835 


97358 


24531 


96945 


i3 


17737 


98414 


19452 


98090 


21161 


97735 


22863 


9735i 


24559 


96987 


47 1 


i4 


17766 


98409 


1 948 1 


98084 


21189 


97729 


22892 


97345 


24587 


96980 


46' 


i5 


17794 


98404 


19509 


98079 


21218 


97723 


22920 


97338 


24615 


96928 


45 


i6 


17823 


gS3gg 


19538 


98073 


21246 


97717 


22948 


973.^1 


24644 


96916 


44 


\l 


17852 


983q4 


19566 


98067 


21275 


97711 


22977 


978^5 


24672 


96909 


43 


17880 


983^^0 


19595 


98061 


2i3o3 


97705 


23oo5 


97318 


24700 


96902 


42; 


19 


17909 


98383 


19623 


98056 


2i33i 


97698 


23o33 


973 1 1 


24728 


96894 


41 j 


20 


17937 


98378 


19652 


98o5o 


2i36o 


97692 


23062 


97304 


24756 


96887 


40^ 


21 


17966 


98373 


19680 


98044 


2i388 


97686 


23090 


97298 


24784 


96880 89 1 


22 


17995 


98368 


19709 


98089 
98033 


21417 


97680 


23ii8 


97291 


24818 


96873 


38; 


23 


18023 


98362 


19737 


21445 


97673 


23146 


97284 


24841 


96866 


37 i 


24 


i8o52 


98357 


19766 


98027 


21474 


97667 


23175 


97278 


24869 


96858 


36 


25 


18081 


98352 


19794 


98021 


2l5o2 


97661 


23203 


97271 


24897 


96851 


35 


26 


18109 


98347 


19823 


98016 


2i53o 


97655 


2323l 


97264 


24925 


96844 


34 


11 


i8i38 


98341 


19851 


98010 


21559 


97648 


28260 


97257 


24953 


96887 


33 


18166 


98336 


19880 


98004 


21587 


97642 


28288 


9725i 


24982 


96829 


32 1 


29 


18195 


98331 


19908 


97998 


21616 


97636 


28816 


97244 


25oio 


96822 


3ii 


30 


18224 


98325 


19937 


97992 


21644 


97630 


28345 


97287 


25o38 


96815 


3o! 


3i 


18252 


98320 


19965 


97987 


21672 


97623 


28878 


97280 


'x5ob6 


96807 


It 


32 


18281 


983 1 5 


19994 


97981 


2I70I 


97617 


28401 


97228 


25094 


9680L 


33 


i83o9 


983x0 


20022 


97975 


21729 


97611 


23429 


97217 


l5l22 


96708 


27 


34 


18338 


983 04 


2oo5i 


97969 


2175s 


97604 


23458 


97210 


25i5i 


96786 


26 


35 


18367 


98299 


20079 


97963 


21786 


97598 


28486 


97208 


25i79 


96778 


25 


36 


18395 


98294 


20108 


97958 


2I8I4 


97592 


285i4 


97196 


20207 


96771 


24 


11 


18424 


98288 


2oi36 


97952 


21843 


97585 


28542 


97189 


25235, 96764 


23 


18452 


98283 


2oi65 


97946 


21871 


97579 


28571 


97182 


25268 96756 


22' 


39 


18481 


98277 


20193 


97940 


21899 


97573 


23599 


97176 


25291 


96749 


^I i 


30 


i85o9 


98272 


20222 


97934 


21928 


97566 


28627 


97169 


25820 


96742 


20' 


41 


i8538 


98267 


20250 


97928 


21956 


97560 


23656 


97162 


25848 


96734 


19! 


42 


18567 


98261 


20279 


97922 


21985 


97553 


28684 


97155 


20876 


96727 


18' 


43 


18595 


98256 


2o3o7 


97916 


220l3 


97547 


28712 


97148 


25404 


96719 


\l 


44 


18624 


98250 


20336 


97910 


22041 


97541 


28740 


97141 


25482 


96712 


45 


18652 


98245 


2o364 


9790D 


22070 


97534 


28769 


97134 


25460 


96705 


i5 


46 


18681 


98240 


20393 


97899 


22098 


97528 


28797 


97127 


25488 


96697 


14 


47 


18710 


98234 


20421 


97893 


22126 


97521 


28825 


97120 


255i6 


96690 


i3 


48 


1*738 


98229 
98223 


20400 


97887 


22l55 


975i5 


28858 


97118 


25545 


96682 


it 


49 


18767 


20478 


97881 


22l83 


97508 


28882 


97106 


25578 


96675 


II 


5o 


18795 


98218 


2o5o7 


97875 


22212 


97502 


28910 


97100 


256oi 


96667 


10 


5i 


18824 


98212 


20535 


97869 


22240 


97496 


289881 97098 


25629 


96660 


9 


52 


18852 


98207 


2o563 


97863 


22268 


97489 


28966 97086 


25657 


96653 





53 


18881 


98201 


20592 


97857 


22297 


97483 


28995 


97079 


25685 


96645 


I 


H 


18910 


98196 


20620 


97851 


22325 


97476 


24028 


97072 


25718 


96688 


55 


18938 


98190 


20649 


97845 


22353 


07470 


24o5i 


97065 


25741 


96680 


5 


56 


18967 


98185 


20677 


97830 
97833 


22382 


97463 


24079 


97o58 


25769 


96623 


4 


u 


18995 


98179 


20706 


22410 


97457 


24108 97o5i 


25798 


96615 


3 


19024 


98174 


20734' 97827 


22438 


97450 


24i36| 97044 


25826, 9660B 


2 


59 

M 


19052 

as. 


98168 


207631 97821 


22467 


97444 


24164 
C. S. 


97087 


25854 
C. S. 


96600 


I 


S. 


C. S. 1 s. 


c. s. 


S. 


1 S. 


1 s. 


"M 




r» Deg. 


78 Deg. 


J 77 ] 


Deg. 


76 Deg. 


76 Deg. 



56 



A TABLE OF NATURAL SINES. 



16 Deg. 



M 



S. 



c. s. 



9 

10 

II 
la 
i3 
14 
i5 

i6 

1:2 

! 19 
30 
21 
22 
23 
24 
25 
26 

II 

?o 



i 3i 

I 33 
I 34 

; 35 

36 

37 
38 
39 
40 
41 
42 
43 
44 
45 

S 

n 

5i 

53 

53 
54 
55 
56 

u 

M 



25882 
25910 
23988 
25966 
25994 
26022 
26o5o 
26079 
26107 
26i35 
26163 
26191 
26219 

26247 
26275 
263o3 

2633i 
26359 
26387 
26415 
26443 

26471 
265oo 
26528 
26556 
26584 
26612 
26640 
26668 
26696 
26724 

26752 
26780 
26808 
26836 
26864 
26892 
26920 
26948 
26976 
27004 
27032 
27060 
27088 
27116 
27144 

• 37172 
27200 
272281 
27256 
37284! 
27312] 
27340 
27368I 
273961 
27424; 
27452 
27480 
27508 
27536 

C. S. 



965o3 
96585 
96378 
96570 
96562 
96555 
96547 
96540 
96532 
96524 
96517 
96509 
96502 
96494 
96486 

96479 
96471 
96463 
96456 
96448 
96440 
96433 
96425 

96417 
96410 
96402 
96394 
96886 
96879 
96871 
96363 

96855 
96847 
96840 
96882 
96324 
96816 
96808 
96801 
96298 
96285 
96277 
96269 
96261 
96253 
96246 

96288 
96280 
96222 
96214 
96206 
96198 
96190 
96182 
96174 
96166 
96x58 
96150 
96142 
96184 



16 Deg. 



74 Dog. 



27564 
27592 
27620 
27648 
27676 
27704 
27781 
27759 
27787 
27815 
27848 
27871 
27899 
27927 
27955 
27988 

28011 
28089 
28067 
28095 
28128 
28i5o 
28178 
28206 
28284 
28262 
28290 
28818 
28846 
28874 
28402 

28429 
28457 
28485 
285i8 
28541 
28569 
28597 
28625 
28652 
28680 



0. S. 



17 Deg. 



C. S. 



18 Deg. 



2 

28786 
28764 
28792 
28820 

28847 
28875 
28908 
28981 
28959 
28987 
29015 
29042 
29070 
29098 
29126 
29154 
29182, 
29209 

C. S. I 



78 Deg. 



96126 1 

96118; 

96110! 

96102 ' 

96094 ' 

96086. 

96078:1 

96070 1 

96062 t 

96054 

9604611 

9608711 

96029 

96021- 

96018 j 

96005 

95997 

95972 

95964 

95956 

95948 

95940! 

95981! 

95928 

95915 

95907 

95898 

958 

958 

95874 
95865 
95857 
95849 
95841 
95882 
95824 
95816 
90807 
95799 

95791 
95782 

95774 
95766 
95757 

95749 
95740 
95782 
95724 
95715 
95707 
95698 
95600 
95681 
95678 
95664 
95656 

95647 
95689I 



29287 
^9265 
29298 
29821 

29848 
29876 
29404 
29482 
29460 
29487 
295i5 
29548 
29571; 
29599 
29626 
29654; 

29682I 

29710 

29787 

29765 

29798 

29821 

29849 

29876 

29904 

29982 

29960 

29987 

8ooi5 

30048 

80071 

30098 
30126 
30154 
30182 
30209 
80287 
80265 
30292 
80820 
30848 
80876 
8o4o3 
30481 
30459 
80486 

8o5i4 
3o542 
80570 
80597 
80625 
3o653 
30680 
80708 
80786 
80768 
80791 
80819 
80846 
80874 ' 
C. S. 1" 



95680 
93622 
93618 
956o5 
95596 
95588 
95579 
95571 
95562 
95554 
95545 
95586 
95528 
95519; 
955111 
95502! 

95408 
954S5 
95476 
95467 
95459 
95450 
95441 
95488 
95424 
95415 
95407 
95808 
95809 
95380 
95872 

95868 
95354 
95845 
95887 
95828 
95819 
95810 
95801 
95298 
95284 
95275 
95266 
95257 
95248 
95240 

95281 
95222 
95213 
95204 
95195 
95186 
95177 
95168 
95159 
95i5o 
95142 
95i38 
95124 
95ii5 



S. 

80902 
80929 
80957 
80985 
81012 
81040 
81068 
8 1 095 
81128 
3ii5i 
81178 
3 1 206 
3i238 
81261 
81289 
3i8i6 

81844 

8187 

81899 

81427 

81454 

81482 

8i5io 

3i587 

3i565 

81598 

31620 

81648 

81675 

81708 

81780 

81758 
81786 
81818 
81841 
81868 
81896 
81928 
81951 
81979 
82006 
82084 
82061 
82089I 
82116; 
82144! 

82171I 

82199 

82227 

82234 

82282 

82809 

82887 

82864 

32892; 

82419 

82447 

32474 

32502 

82529 

"cTsT' 



c. s. 



19 Dog. 

S. I s.c. 



M 



93097; 



95079 
95070 
95061 
95o52 
95043 
95o83 
95024 
95oi5; 
95oq6; 

949971: 
94988, 

9497911 

94970 

94961 il 

94952 

94948 

94988 

94924 

94915 

94906 

948, 

94888 

94878 

94869 

94860 

9485 1 

94842 

94882 

94828 
94814 
94805 

94795 
94786 

94777 
94768 
94758 
94749 
94740 
94780 
94721 
94712 
94702 
94698 

94684 
94674 
94665 
94656 
94646 
94687 
94627 
94618 
94609 
94399 
94590 
94580 
94571 
94561 

S. 



32557 
32584' 
82612, 
82689 
82667 
82694 
82722 
82749 

32777! 
82804; 
82832i 
32859 
82887 
82914 
82942 
82969 

32997 

880241 

33o5i 

88079! 

83io6! 

38i34 

83i6i 

88189 

83216 

88244 

88271 

88298 

33826 

88858 

88881 

88408 
33436 
33468 
88490 
835i8 
33545 
33573 
88600 
88627 
38655 
88682 
88710 
38787 
88764 
88792 

88819 
88846 
88874 
88901' 
33929' 
88956! 
88983, 
3401 1, 
34088 
84065 
84098 
34120 

84147 
34175 

■cTs: ■ 



94552 
94542 
94533 






94523; 57! 

94514: 56 • 

94504I 55' 

94495; 54! 

94485 53 

94476 

94466 

94457 

94447 

94488 

94428 

94418 

94409 



52 

5i| 
5o 

tv, 

47 I 

46 
45 

441 

43; 

421 
41 1 
40 j 

III 

36 



94899 

94800 

94380 

94870 

94861 

94351 

94842 

94882 

94822 

94818] 35 

94808 

94298 

94284 

94274 

94264 

94254 
94245 
94235 
94225 
94215 
94206 

94196 
94186 

94176 
94167 
94157 
94147 
94187 

94127 

94II8 

94108 14 
94098 1 3 

$4088; 13 
94078 II 
94068] ID 

94o58 
94049 
94089 
94029 
94019 
94009 
98999 
98989 
93979 

S. 



34 
33 

32 

3i 
3o 

25 

24 

23 
22 
21 
20 
I( 

il 

i5 



li 



72 Deg. 



71 Deg. 



70 Deg. 



A TABLE OF NATURAL SINES. 



dl 



M 


20 Deg. 


21 Deg. 


22 


Deg. 


23 Deg. 


r 24 


Deg. 


M 

60 


S. 


C. S. 


S. 


c. s. 
98358 


S. I c. s. 


S. 
89078 


C. S. 


r s. 


TTsT 
91355 





34202 


93969 


i 35837 


37461 92718 


92o5o [ 40674 


I 


34229 


93959'; 35864 


98848 


87488 92707 


89100 92089 


40700 


91843 


U 


2 


34257 


93949 35891 


98887 


375i5 92607 


89127 


92028 


40727 


9i33i 


3 


34284 


93939 


! 30918 


98827 


87542 92686 


89153 


92016 


40753 91819 


571 


4 


343 1 1 


98929 


35945 


98816 


87569 


91675 


89180 


92005 


40780 


91807 


56 


5 


34339 


03919 


35978 


98806 


87595 


92 64 


39207 


91904 


40806 


91295 


55 


6 ! 34366 


98909 


86000 


98295 


37022 92053 


89284 


91982 


40833 


91288 


54 


I 


34393 


93899 


36027 


98285 


87649 


92642 


1 89260 


91971 


40860 


91272' 53 


34421 


93889 


36o54 


98274 


87676 


92681 


89287 


91900 


40886 


91260; 52 


9 


34448 


93879 


36081 


98264 


87703 


92620 


89814 


91948 


40918 


91248 


5i 


ic 


34475 


93869 


86108 


98253 


87780 


9260Q 


39841 


91986 


40989 


91286 


5o 


,11 


345o3 


98859 


86185 


98248 


87757 


92508 


89867 


91925 


40966 


01924 


49 


12 


3453o 


98849 


86162 


98282 


37784 


92587 


89894 


91914 


40992 


91212 


48 


i3 


34557 


98889 


36190 


98222 


87811 


92576 


89421 


91902 


41019 


91200 


47 


14 


34584 


98829 


36217 


982II 


87888 


92565 


89448 


91891 


41045 


91188 


46 


i5 


34612 


93819 


36244 


98201 


87865 


92554 


89474 


91879 


41072 


9II76 


45 


i6 


34639 


98809 


36271 


98190 


87892 


92543 


89501 


91868 


41098 


9I164 


44 


\l 


34666 


98709 


36298 


93180 


37919 


92582 


89528 


9i856 


41125 


9ii52 


43 


34694 


98789 


36825 


98169 


87946 


92521 


89555 


91845 


4ii5i 


91140 


42 


19 


34721 


9^779 


36852 


98159 


87978 


92510 


89581 


91888 


41178 


91128 


41 


20 


34748 


98769 


86879 


98148 


37999 


92499 


89608 


91822 


41204 


91116 


40 


21 


34775 


98750 


36406 


98187 


88026 


92488 


89635 


91810 


41281 


91104 


39 


22 


34803 


98748 


36434 


98127 


38o53 


92477 


89661 


91799 


41257 


91092 


38 


23 


34830 


98788 


86461 


98116 


88080 


92466 


89688 


91787 


41284 


91080 


37 


24 


34857 


98728 


36488 


98106 


88107 


92455 


89715 


91775 


41810 


91068 


36 


25 


34884 


98718 


365i5 98095 


88184 


92444 


89741 


91764 


41887 


9io56 


35 


26 


34912 


98708 


36542 


98084 


38i6i 


92482 


39768 


91752 


41868 


91044 


34 


27 34939 


98698 


86569 


98074 


88188 


92421 


89795 


91741 


41890 


91082 


33 


28 1 34966 


93688 


86596 


98063 


382x5 


92410 


89822 


91729 


41416 


91020 


32 


29 


34993 


98677 


86628 


98052 


88241 


92890 


89848 


91718 


41448 


91008 


81 


So 


35c2i 


98667 


86650 


98042 


88268 


92888 


89875 


91706 


41469 


90996 


3o 


3i 


35048 


98657 


86677 


98081 


88295 


92877 


89902 


91694 


41496 


90984 


^2 


32 


35075 


98647 


86704 


98020 


88822 


92866 


89928 


91688 


4l522 


90972 


28 


33 


35io2 


98687 


86781 


98010 


88849 


92355 


89955 


91671 


41549 


90960 


27 


34 


35i3o 


98626 


86758 


92999 


88876 


92843 


89982 


91660 


41575 


90948 


26 


35 


35i57 


98616 


86785 92988I 


88408 


92882 


40008 


91648 


41602 


90986 


25 


36 


35i83 


98606 


36812 


92978 


88480 


92821 


4008 5 


91686 


41628 


90924 


24 


37 


352II 


98596 


86889 


92967 


88456 


92810 


40062 


91625 


41655 


90911 


33 


38 


35239 


98585 


36867 


92956 


38488 


92299 


40088 


91618 


4I68I 


90899 


22 


39 


35266 


93575 


86894 


92945 


385io 


92287 


4oii5 


91601 


41707 


90881 


ai 


40 


35293 


98565 


86921 


92985 


88587 


92276 


4oi4i 


91590 


41734 


90875 


20 


41 


35320 


98555 


86948 


92924 


38564 


92265 


40168 


91578 


41760 


90868 


IQ 


42 


35347 


98544 


86975 


92918 


88591 


92254 


40195 


91566 


41787 


9085 1 


18 


43 


35375 


98534 


87002 


92902 


88617 


92243 


40221 


91555 


41818 


90889 


\l 


44 


35402 


93524 


87029 


92892 


38644 


92281 


40248 


91548 


41840 


90826 


45 


35429 


93514 


37056 


92881 


88671 


92220 


40275 


9i58i 


41866 


90814 


i5 


46 


35456 


93508 


87083 


92870 


88698 


9220Q 


40801 


91519 


41892 


90802 


14 


47 


35484 


98498 


87110 


92859 


38725 


921Q8 


40828 


9i5o8 


4i9'9 


90790 


i3 


48 


355ii 


98488 


87187 


92849 


38752 


92186 


4o855 


91496 


41945 


90778 


12 


^9 


35538 


93472 


87164 


92888 


38778 


92175 


40881 


91484 


41972 


90766 


II 


5o 


35565 


98462 


87191 


92827 


388o5 


92164 


40408 


91472 


41998 


90753 


10 


5i 


3559^ 


98451 


87218 


92816 


88882 


92152 


40484 


91461 


42024 


90741 


q 


55 


35619 


93441 


37245 


92805 


88859 


92141 


40461 


91449 


42o5i 


90729 





53 


35647 


98431 


87272 


92794 


88886 


92180 


40488 


91487 


42077 


90717 


7 


54 


35674 


98420 


87299 92784! 


38912 


92119 


4o5i4 


91425 


42104 


90704 


6 


55 


35701 


98410 


37826} 927781 


38989 


92107 


40541 


91414 


42180 


90692 


5 


56 


357:8 


98400 


873581 92762 


88966 


92096 


4o567 


91402 


42 1 56 


90680 


4 


57 


35755 


98889 


87880 


92751 


88998 


92085 


40594 


91890 


42183 


90668 


3 


58 i 35782 


98870 


87407 


92740 


89020 


92078 


40621 


91878 


42209 
42235 


90655 


2 


59 


35oio 


98868 


37434 


92729 


39046 


92062 


40647 

C. S. 


91866 

S. 


90643 

S. 


I 


M 


C. S. 


S. 


c. s. i s. 


C. S. 


S. 


"cTsT 


I 


69 Beg. 


68 Deg. 


67 Deg 1 


66 Deg. 


66 Deur. 



19 



68 



A TABLE OF NATURAL SINES. 



M 


25 J 


Dfcg. 


26 Deg. 


27 Deg. 


28 Deg. 


29 Deg. 


1 


S 


C. S. 
9063 1 


S. 


c. s. 


S. 1 c. s. 


S. 


C. S. 
88295 


s. 1 c. s. 





42262 


43837 


898791! 45399 89IOI 


! 46947 


48481' 87462 60 


I 


42288 


906x8 


43863 


89867!! 45425 89087 i 46973 


8828X 


485o6 87448 59 


3 


423i5 


90606 


43889 


89804 


4545 X 89074 46999 


88267 


48532 87434 


58 


3 


42341 


Q0594 


43916 


8984X 


45477 89061 47024 


88254 


48557 87420 


5- 


4 


42367 


90082 


43942 


89828^1 455o3 89048 ' 47o5o 


88240 


48583! 87406 >^ 


5 


42394 


90069 


43968 


898161! 45529 


89035 ! 47076 


88226 


48608 8739 X 55 


6 


42420 


90557 


43994 


89803'! 45554 


89021 471OX 


882x3 


48634 


87377 54 


7 


42446 


90045 


44020 


89790,1 45580 


89008 47127 


88199 


48659 


87363 53 


8 


42473' 90532 


44046 


89777J 45606 


88995 47153 


88x85 


48684 


87349 5j 


9 


42499 90520 


44072 


89764 45632 


88981;; 47178 


88x72 


48710 


87335 


5i 


iO 


42525 


90507 


44098 


89752II 45658 


88968 ' 47204 


88x58 


48735 


87321 


5o 


II 


42552 


90495 


44124 


89739 


45684 


88955:! 47229 


88144 


4876X 


87306 


49 


12 


42578 


90483 


44i5x 


89726 


45710 


88942'! 47200 


88i3oi| 48786 


87292 


48 


i3 


42604 


90470 


44177 


897x3 


45736 


88928'; 47281 


881x7 


48811 


87278 


47 


14 


4263 1 


90458 


442o3 


89700 


45762 


88915, 47306 


88xo3 


48837 


87264 


46 


i5 


42657 


90446 


44229 


89687 


45787 


88902 47332 


88089 


48862 


87250 


45 


i6 1 4 2683 


90433 


44255 


89674 


458x3 


88888 47358 


88075 


48888 


87235 


44 


\l 


42709 


90421 


44281 


89662 


45839 


88875! 47383 


88062 


48qi3 


8722X 


43 


42736 


90408 


44307 


89649' 


45865 


88862'! 47409 


88048 


48938 


87207 


42 


39 


42762 


90396 


44333 


89636' 


45891 


88848,1 47434 


88034 


48964 


87193 


41 


20 


42788 


9o3»3 


44359 


89623 


45917 


88835 


47460 


88020 


48989 


87178 


40 


21 


428x5 


90371 


44380 


89610! 


45942 


88822 


47486 


88006 


490x4 


87164 


39 

38 


22 


42841 


90358 


44411 


89O97! 


45968 


88808 


47511. 87993 


49040 


87i5o 


23 


42867 


90346 


44437 


89084! 


45994 


88795 


47537; 87979 


49065 


87136 


37 


24 


42894 


90334 


44464 


89571 


46020 


88782!! 47562 


87965 


49090 


87x21 


36 


25 


42920 


903 2 X 


44490 


89558 


46046 


88768 


47588 


87951 


49116 


87107 


35 


26 


42946 


90309 


445x6 


895451 


46072 


88755 


47(514 


87937 


49141 


87093 


34 


27 


42972 


90296 


44542 


890321 


46097 


88741 


47639 


87923 


49166 


87079 


33 


28 


42999 


90284 


44568 


89519 


46123 


88728 


47665 


87909 


49192 


87064 


32 


29 


43025 


90271 


44594 


89506 


46149 


887x5 


47690 


87896 


49217 


87o5o 


3i 


3o 


43o5i 


90259 


44620 


89493 


46x75 


8870X 


47716 


87882 


49242 


87036 


3o 


3i 


43077 


90246 


44646 


89480 


4620X 


88688 


47741 


87868 


49268 


87021 


29 


32 


43x04 


90233 


44672 


89467 


46226 


88674 


47767 


87854 


49293 


87007 


28 


33 


43i3o 


90221 


44698 


89454! 


46202 


88661 


47793 


87840 


49318 


86993 


27 


34 


43 1 56 


90208 


44724 


89441!! 46278 


88647 


478x8 


87826 


49344 


86978 


26 


35 


43x82 


90x96 


447^0 


89428 


46304 


88634! 


47844 


878x2 


49369 


86964 


25 


36 


43209 


90183 


44776 


89415 


46330 


88620!! 47869 


87798 


49394 


86949 


24 


37 


4323 d 


9017X 


44802 


89402 46355 


88607 


47895 


87784 


494x9 


86935 


23 


38 


43261 


90x58 


44828 


89389I 46381 


88593 


47920 


87770! 


49445 


86921 


22 


39 


43287 


90x46 


44854 


89376I 


46407 


8858oi 


47946 


87756 


49470 


86906 


2X 


40 


433x3' 


90x33 


44880 


89363 


46433 


88566! 


47971 87743 


49495 


86892 


20 


41 


43340 


90120 


44906 


89350 


46458 


88553: 


47997 87729 


4952 X 


86878 


XO 


42 


43366 


90x08 


44932 


89337 


46484 


88539 


48022 877x5 


49546 


86863 


x8 


43 


43392 


90095 


449 08 


89324 


465x0 


88526 


48048 87701 


49571 


86849 


17 


44 


43418 


90082 


44984 893 II 1 


46036 


885x2, 


48073 


87687 


49596 


86834 


16 


45 


43445 


90070 


45oio 


89298 


4656 X 


88499! 


48099 


87673 


49622 


86820 


i5 


46 


43471 


90057 


45o36 


89285 


46587 


88485 


48x24 


87659 


49647 


868o5 


14 


^J 


43497 


90045 


45062 


89272 


466x3 


884721! 48x5o 876451 


49672 


86191 


i3 


4a 


43523 


90032 


45o88 


89259 


4663g 


88458 j 


48x75 8763X 


49697 


86777. [2| 


49 


43549 


900x9 


45x14 


89245 


46664 


88445 


48201 87617 


49723 86762! II 


56 


43575 


00007 


45x40 


89232 


46690 


8843 X 


48226 87603 


49748 1 86748 


5i 


43602 


45x66 


892x9 


467x6 


88417 


48252 87589 


49773 86733- ol 


52 


43628 


8998 X 


45x92 


89206 


46742 


88404' 


48277 


87570 4Q798 867x9 


S 


53 


43654 


89968 


452x8 


89193 


46767 


88390 


483o3 


3756x!! 4q824, 86704 


I 


04 


43680 


89956 


45243 


89x80 


46793 


88377 


48328 


87546 ' 40849* 86690 


55 


43706 


89943 


40269 


89167 


468x9 


88363 


48354 


87532 


49874 86675 


5 


56 


43733 


89930 


45295 


891531 


46844 


88349 


48379 


875x8; 


40899' 86661 


4 


57 
■ 5S 


43759 


899x81 


453 2 X 


89140; 


46870 


88336 


48400 


87504^ 


40924 86646 


3 


43i85 


89905 


45347 


89127' 


46896 88322! 


48430 


87490, 


4q95o, 86632 


X 


59 


438x1 


89892 


45373 


89114' 


4692 x' 883o8j 


48456 


87476 


49975 86617 

C. S. 1 s. 


I 
M 


1 M 


e.g. 


S. 


C. S. 


s. ! 


c. s. . s. 


C. S. 


S. 


1 


04] 


>eg. 


i 08 I 


^i._l 


62 I 


^cg. J 


611 


)e^. 


6:> Dog. 



A TABLE OF NATURAL SINES. 



69 



M 


80 ] 


Deg. 

~C. S. 


31 : 


Deg. 


32 Deg. 


83 Deg. 


84 Deg. 


' 

M 

60 


S. 


S. 


C. S. 
857.7 


S. 
52992 


C. S. 


S. 1 c. s 
54464 83867 


S. 


c. s. 
82904 


o 


5oooo 


866o3 


5i5o4 


84806 


569x9 


I 


5oo25 


86588 


5x529 


83702 


53017 84789 


54488 8385x 66943, 82887 


5o 


2 


5oo5o 


86573 


5i554 


83687 


53o4i 84774 


545i3 83836 66968! 82871' 58 


3 


50076 


86559 


5 1 579 


85672 


53o66 84769 


54537' 83819 


■J6992! 82866 


57 


4 


5oioi 


86544 


5 1 604 


85657 


5309 X 84743 


54661 838o4 


66016! 82839 


56 


5 


50126 


86530 


51628 


85642 


53xx5, 84728 


54686! 83788 


5604c 82822 


55 


6 


5oi5i 


865 1 5 


5x653 


85627 


53 1 4c 


84712 


54610 83772 


66064 


82806 


54 


6 


50176 


865ox 


51678 


85612 


53x64 


84697 


54636 83766 


56o88 


82790 


53 


5020I 


86486 


51703 


85597 


53189 


8468 X 


5463g 83740 


56xx2 


82773 


52 


9 


50227 


86471 


5x728 


85582 


532X4 


84666 


54683 83724 


56x36 


82767 


5i 


lO 


50252 


86457 


5x753 


85567 


53238 


84660 


54708 


83708 


66160 


82741 


5o 


II 


50277 


86442 


5x778 


85531 


53263 


84635 


54732 


83692 


66x84 


82724 


49 


12 


5o3o2 


86427 


5i8o3 


85536 


53288 


84619 


64766 


83676 


66208 


82708 


48 


i3 


5o327 


86413 


5x828 


85521 


533x2 


84604 


5478 X 


83660 


56232 


82692 


47 


i4 


5o352 


86398 


5x852 


855o6 


53337 


84688 


64806 


83645 


66266 


82676 


46 


i5 


5o377 


86384 


51877 


85491 


5336X 


84673 


54829 


83629 


56280 


82669 


45 


i6 


5o4o3 


86369 


51902 


85476 


53386 


84667 


64864 


836 1 3 


563o5 


82643 


44 


\l 


50428 


86354 


5x927 


85461 


5341 1 


84642 


54878 


83697 


56329 


82626 


43 


50453 


86340 


5x952 


85446 


53435 


84526 


54902 


8368i 


56353 


82610 


42 


19 


50478 


86325 


5x977 


8543 1 


53460 


845 XX 


54927 


83665 


56377 


82693 


41 


20 


5o5o3 


863x0 


52002 


83416 


53484 


84495 


6496 X 


83649 


6640 X 


82677 


40 


21 


5o528 


86295 


52026 


85401 


53509 


84480 


64976 


83633 


66426 


8266X 


39 


22 


5o553 


8628X 


52o5i 


85385 


53534 


84464 


54999 


836i7 


66449 


82644 


38 


23 


50578 


86266 


52076 


85370 


53558 


84448 


66024 


836oi 


66473 


82628 


37 


24 


5o6o3 


8625x 


52X01 


85355 


53583 


84433 


55048 


83485 


56497 


826x1 


36 


25 


50628 


86237 


52126 


85340 


53607 


84417 


66072 


83469 


56621 


82496 


35 


26 


5o654 


86222 


52x5x 


85323 


53632 


84402 


66097 


83453 


66646 


82478 


34 


^1 


50679 


86207 


52x75 


85310 


53656 


84386 


55i2x 


83437 


66669 


82462 


33 


28 


50704 


86192 


52200 


85294 


5368x 


84370 


55x45 


83421 


66693 


82446 


32 


29 


50729 


86178 


52225 


85279 


53706 


84356 


66169 


834o5 


666x7 


82429 


3x 


3o 


50754 


86x63 


5225o 


85264 


53730 


84339 


66x94 


83389 


5664X 


82413 


3o; 


3i 


50779 


86x48 


52275 


85249 


53764 


84324 


562x8 


83373 


66665 


82396 


29 


32 


5o8o4 


86x33 


52299 


85234 


53779 


84308 


66242 


83366 


66689 


82380 


28 


33 


50829 


86119 


52324 


852x8 


53804 


84292 


66266 


83340 


66713 


82363 


27 


34 


5o854 


86104 


52349 


85203 


53828 


84277 


66291 


83324 


66736 


82347 


26 


35 


50879 


86089 


52374 


85x88 


53853 


8426X 


553i6 


833o8 


56760 


82330 


25 


36 


50904 


86074 


52399 


85x73 


53877 


84246 


65339 


83292 


66784 


823x4 


24 


H 


50929 


86069 


52423 


85x57 


53902 


84230 


55363 


83276 


66808 


82297 


23 


38 


50934 


86045 


52448 


85x42 


53926 


84214 


56388 


83260 


56832 


82281 


22 


39 


50979 


86o3o 


52473 


85x27 


5393X 


84198 


66412 


83244 


66856 


82264 


2X 


40 


5 1 004 


8601 5 


52498 


85ix2 


53976 


84182 


66436 


83228 


66880 


82248 


20 


41 


51029 


86000 


52522 


86096 


64000 


84167 


66460 


832X2 


66904 


8223l 


XO 


42 


5io54 


83985 


52547 


86081 


54024 


841 5 1 


55484 


83x96 


66928 


822x4 


18 


43 


5io79 


85970 


52572 


86066 


64049 


84i35 


55509 


83x79 


66962 


82x98 


nt 


44 


5x104 


83966 


52697 


85o5i 


54073 


84120 


56533 


83x63 


56976 


82x81 


i6j 


45 


5x1 29 


85941 


52621 


85o35 


64097 


84104 


55557 


83x47 


67000 


82165 


i5\ 


46 


5ix54 


85926 


52646 


86020 


54x22 


84088 


6658x 


83x3i 


57024 


8ii48 


ui 


47 


5xx79 


8391 X 


52671 


86oo5 


54146 


84072 


556o6 


83xi5 


57047 


82l32 


i3 


48 


5 1 204 


85896 


52696 


84989 


54171 


84067 


55630 


83098 
83o82 


57071 


82x16 


12 


P 


5X229 


8588i 


52720 


84974 


54195 


84041 


55664 


57096 


82098 


II 


5o 


5x254 


85866 


52745 


84969 


54220 


84026 


66678 


83o66 


57119 
57143 


82082 


XO 


5i 


5x279 


8585x 


52770 


84943 


54244 


84009 


66702 


83o6o 


82065 





52 


5i3o4 


85836 


52794 


84928 


54269 


83994 


66726 


83o34 


57167 


82048 





53 


5x329 


8582X 


528x9 


84913 


54293 


83978 


56760 


83ox7 


67x91 


82032 


7 


54 


5x354 


858o6 


52844 


84897 


54317 


83962 


66776 


83oox 


672x6 


820x5 


6 


55 


5x379 


85792 


52869 


84882 


54342 


83946 


66790 
55823 


82986 


67238 


8x999 
81982 


5 


56 


5 1 404 


85777 


52893 


84866 


54366 


83930 


82969 


67262 


4 


U 


51429 


85762 


52918 


8486 X 


5439X 


839x5 


66847 


82953 


67286 


81965 


3 


51454 


85747 


52943 


84836 


64416 


83899 
83883 

S. 


5587 X 


82936 


67310 


81949 


2 


§9 1 5i479 


85732 


52967 


84820 


5/i/|/io 


55896 
C. S. 


82920 


67334 

C. S. 


81932 


I 
M 

1 


M 


C. S. 


S. 


C. S. 
58 I 


S. 


C. S. 


S. 


S. 




69 I 


)e^ 


57 De^. 1 


56 Deg. 6 


55 Deg. ! 



70 



A TABLE OF NATURAL SINES. 



85 


Deg. 


86 Deg. 


37 Deg. 


38 Deg. 


89 Deg. 




M J^' " 


: C. S. 


S. 
58779 


! C. S. 


S. 


C. S. 


S. 


C. S. 


S. 


C. S. 


M 





57358 


81915 


80902 


60182 


79864 


61666 


78801 


6':932 


77715 


bo 


I 


57381 1 81899 


588o2 


80886 


60206 


79846 


61689 


78788 


62966 


77696 


^ 


2 


57405 81882 


58826 


80867 


60228 


79829 


61612 


78766 


i 62977 


77678 


3 


57429! 8 I 865 
57453 1 81848 


58849 


80860 


60261 


79811 


61686 


78747 


68000 


77660 67 


4 


58873 


80833 


60274 


79793 


61668 


78729 


1 68022 


77641, 56 


5 


57477 8i832 


58896 


80816 


60298 


79776 


61681 


7871 1 


68046 


77623 


55 


t 


57501 


1 8i8i5 


58920 


80799 


60821 


79768 


61704 


78694 


68068 


77606 


54 


% 


57524 


' 81708 


58943 


80782 


60844 


79741 


61726 


78676 


68090 


77686 


53 


8 


57548 


i 81782 


58967 


80765 


60867 


79728 


61749 


78668 


63ii3 


77668 


52 


9 


57572 


1 81765 


58990 


80748 


60890 


79706 


61772 


78640 


68i36' 77660 


5i 


IC 


57596 


i 81748 


59014 


80730 


60414 


79688 


61795 


78622 


68168 


77531 


5o 


II 


57619 


1 81731 


59037 


80713 


60487 


79671 


61818 


78604 


68180 


77613 


49 


13 


57643 


i 81714 


59061 


80696 


60460 


79668 


61841 


78686 


632o3 


77494 


48 


i3 


57667 


81698 


59084 


80679 


60488 


79636 


61864 


78668 


63226 


i 77476 


47 


14 


57691 


81681 


59108 


8^632 


60606 


79618 


61887 


78660 


68248 


77468 


46 


i5 


57715 


81664 


59131 


80644 


60629 


79600 


61909 


78682 


68271 


77439 


46 


i6 


57738 


81647 


69154 


80627 


6o553 


79688 


61982 


78614 


63293 


77421 


44 


\l 


57762 


8i63i 


59178 


80610 


60676 


79666 


61966 


78496 


633 1 6 


77402 


43 


57786 


81614 


59201 


80693 


60699 


79547 


61978 


78478 


65338 


77384 


42 


19 


57810 


81 597 


59225 


80676 


60622 


79680 


62001 


78460 


63361 


77866 


41 


20 


57833 


8i58o 


59248 


80668 


60646 


79612 


62024 


78442 


68383 


77347 


40 


21 


57857 


8 1 563 


59272 


80641 


60668 


79494 


62046 


78424 


68406 


77829 


39 


22 


57881 


81546 


59295 


80624 


60691 


79477 


62069 


78406 


68428 


77810 


38 


23 


57904 


8i53o 


59318 


80607 


60714 


79469 


62092 


78887 


68461 


77292 


37 


24 


57928 


8i5i3 


59842 


80489 


60788 


79441 


62116 


78869 


68473 


77273 


36 


25 


57952 


81496 


59365 


80472 


60761 


79424 


62i38 


78861 


68496 


77266 


85 


26 


57976 


81479 


59389 


80466 


60784 


79406 


62160 


78888 


68618 


77286 


34 


13 


57999 


81462 


59412 


80438 


60807 


79888 


62183 


78816 


68640 


77218 


33 


58023 


81445 


59436 


80420 


6o83o 


79871 


62206 


78297 


68663 


77199 


32 


29 


58047 


81428 


59459 


80403 


60868 


79363 


62229 


78279 


68685 


77181 


3i 


3o 


58070 


81412 


59482 


8o386 


60876 


79886 


62261 


78261 


68608 


77162 


3o 


3i 


58094 


81395 


59606 


8o368 


60899 


79818 


62274 


78243 


68o3o 


77144 


29 


32 


58ii8 


81378 


69629 


8o36i 


60922 


79800 


62297 


78226 


63663 


77126 


28 


33 


58i4i 


8i36i 


59662 


80334 


60946 


79282 


62820 


78206 


68676 


77107 


27 


34 


58i65 


81 344 


69676 


8o3i6 


60968 


79264 


62342 


78188 


68698 


77088 


26 


35 


58189 


81327 


69699 


80299 


60991 


79247 


62866 


78170 


68720 


77070 


25 


36 


58212 


8i3io 


69622 


80282 


61016 


79229 


62888 


78162 


68742 


77061 


24 


37 


58236 


81293 


69646 


80264 


6io38 


7921 1 


62411 


78184 


68766 


77088 


23 


38 


58260 


81276 


69669 


80247 


61061 


79198 


62433 


78116 


68787 


77014 


»a 


39 


58283 


81259 


59693 


80280 


61084 


79176 


62466I 


78098 


68810 


76996 


21 


40 


583o7 


81242 


69716 


80212 


61107 


79168 


62479 


78079 


68832 


76977 


20 i 


41 


58330 


81225 


59739 


80196 


61180 


79140 


62602 


78061 


68854 


76969 19 


42 


58354 


81208 


69763 


80178 


6u53 


79122 


62624 


78043 


68877 


76940 18 


43 


58378 


81191 


69786 


80160 


61176 


79106 


62647 1 


78026 


68899 


769211 17 


44 


58401 


81174 


59609 


80143 


61 199 


79087 


62670 


78007 


68922 


76908 


16 


45 


58425 


8ii5'7 


69832 


80126 


61222 


79069 


62692: 


77988 


68944 


76884 


i5 


4& 


58449 


8.140 


69866 


80108 


61245 


79061 


62616! 


77970 


68966 


76866 


14 


^J ' 


58472 


81123 


59879 


80091 


61268 


7go33 


62688! 


77962 


63989 


76847 


i3 


i 48 ' 58496' 81106 


69902, 80073 


61291 


79016 


62660 


77934 


640 II 


76828 


12 


1 ir , 585i9 81089 


69926 80066 


61814 


78998 


62688^ 


77916 


64033 


76810 11 


1 5c ; 58543 1 


81072 
3io55 


69949 8oo38 


618871 


78980 


62706 


77897 


64066 


76791 10 


5i i 58567 


O9972 8oG2I 


61860' 


78962 


62728 


77879 


64078 


76772, 9 


5a 1 53590 


8io38 


69995 


8ooo3 


61888 


78944 


62751I 


77861 


64 1 00 1 76764 


8 


53 


58614 


81021 


60019 


79986 


61406 


78926 


627741 


77843 


641 281 76735 


I 


54 


53637 


81004 


60042 


79968 


61429 


78908 


62796 


77824 


64145 


76717 
76698 


55 


5866 1 


80987 


60066 


79961 


61461 


78891 


62819 


77806 


64167 


5 


56 


58684 


80970 


60089 799^4 


61474 


78873 


62842 


77788 


64 I 901 76679 


4 


ll 


58708 


80953 


601 1 2 79916 


61497 


78855 


62864 


77769 


642121 76661 


3 


58731 


80936 


6oi36| 


79809 


61620 


78887 


62887 


77761 


64284 76642 


2 


59 
M 


58755 


80919 


60168 

C. S. 


79881 


61643 


78819 


62909 


77733 

s. 


64266! 76623 


I 
M 

1 


C. S. 1 


S. 


S. 


C. S. 


S. 


C. S. 


C. S. ! S. 


54 Deg 


53 Deg. i 


52 Deg. 1 


51 Deg. 1 


50 Deg. 



A TABLE OF J!^ATURAL SINES. 



n 



9 

!0 
II 

la 
i3 
M 
i5 

i6 

i8 

20 

31 
33 
23 
24 

35 
26 
37 
38 
39 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 
40 
41 
42 
43 
44 
45 

46 

47 
<8 

5i 

52 

53 

54 
55 
56 

U 

59 

M 



40 I>eg. 

8. C. S. 



7615 

76135 

76116 

76097 

76078 

76o5g 



64379 
64301 
64333 
64346 
64368: 
643 90 I 
64413 
64435' 
64457] 

64479 
64001 j 
64524' 
64546! 
64568 1 
64590 
6461 2 1 

64635 

64657 

64679 

64701 

64723, 

64746 

64768^ 

64790 

64812! 

64834' 

64856 

64878; 

64901! 

64923 

64945 J 

64967 

64989' 

65oii 

65o33 

65o55 

65o77 

65o99 

65l22 

65 1 44 
65i66, 
65 1 88 
65210 
65232 
65254 
65276 

65298 
65320 
65342 
65364 
65386 
654o8 
6543o 
65452 
65474 
65496 
655i8 
65540 
65562 
65584 
656o6 

48 Deg. 



766<y4 
76586 
76567 
76548 
76530 
76511 



41 Deg. 



C. S. 



656o6 
65628! 
6565o' 
65672! 
65694 
65716 



764921 65738 
76473 I 65759! 
76455!! 65781 [ 
76436 1| 658o3l 
764171! 65825| 



76398 
76380 
76361 
76342 
76323 

76304 



65847 
65869' 
65891 j 
65913! 
65935: 

65956! 



76286I; 65978 
76267!! 66000 
76248]! 66022 
76229!! 66044 
76210,1 66066 
76192!! 66088 
76173!! 66109 



66i3i 
661 53 
66175 
66197 
66218 
66240 



76041!! 66262 
76022!' 66284 



76003 
75984 
73965 
75946 
75927 
75908 
75889 
75870 
7585i 
75832 
758i3 



663o6 
66327 
66349 
66371 
66393 
66414 
66436 
66458! 
66480 
665oi 
66523 



75794 66545 
75775!! 66566 



75756 

75738 

75719 
75699 
75680 
75661 
75642 



66588 

66610 
66632 
66653 
66675 
66697 
66718 



75623.' 66740 
75604!! 66762 
75585ll 66783 
75566!j 668o5 
75547 I 66827 
75528! 66848 
75509 I 66870 
75490 I 66891 
75471 ii 66913 

"cTsT 



75471 
75452 
75433 
75414 
75395 
75375 

75356 

75337 

75318 

75209 

75280 

75261 

75241 

752221 

75203I 

751841 

75165' 

75146 

75126 

75107 

75088 

75069 

75o5o 

75o3o 

75oii 

74992 

74973 

74953 

74934 

740 1 5 

74S96 
74876 
74857 
74838 
74818 
74799 

74760 

74741 

74722 

74703 

74683 

74664 

746 

74625 

74606 

74586 
74567 
74548 
74528 
74509 
74489 
74470 
74451 
74431 
74412 
74392 
74373 
74353 
74334 
74314 



48 Deg. 



42 Deg. 



S. 



66913 
66935 
66956 
66978 
66999 
67021 
67043 
67064 
67086 
67107 
67129 
67151 
67172 
67194 
672151 
67237! 

67258 
67280 
67301 
67323 
67344 
67366 
67387 
67409 
67430 
67452 
67473 
67495 
67516 
67538 
67559 

67580 
67602 
67623 
67645 
67666 
67688 
67709 
67730 
67752 
67773 
67795 
67816 
67837 
67859 
67880 

67901 
67923 
67944 
67965 

67987 
6^008 
68029 
68o5i| 
68072! 
680931 
68ii5| 
68i36! 
68157I 
68179! 
68200! 

cTsTI 



c. s. 



74314 
74295 
74276 

74256 
74237 
74217 
74198 
74178 
74159 
74139 
74120 
74100 
74080 
74061 
74041 
74022 

74002 
73983 
73963 
73944! 
73924' 
73904 
73885 
73865 
73846 
73826 
73806! 

73787 
73767 

73747 
73728 

73708, 

73688 

73669 

73649' 

73629! 

73610J 

73590, 

73570 

73551I 

73531 1 

73511 

73491! 

734721 

73452! 

7343 2 j 

73412! 
733931 
73373 
73353! 
73333I 
73314' 
73294 
73274 
73254 
73234 
732i5i 
73195I 
73175; 
73i55! 
73i35 

S. 
47 Deg. _ 



48 Deg. 



C. S. 



6820c. 
68221 
68242 
68264 
68285 
683o6 
68327 
68349 
68370 
68391 
68412 
68433 
68455 
68476 
68497 
685 1 8 

68539 
6856i 
68582 
686o3 
68624 
68645 
68666 
68688 
68709 
68730 
68751 
68772 
68793 
68814 
68835 

68857 



68899 
68920 
68941 
68962 
68983 
69004 
69025 
69046 
69067 
69088 
69109 
69130 
6qi5i 

69172 

69193 

69214 

69235! 

69256! 

69277' 

69298 

69319 

69340 

69361 

69382 

69403 

69424 

69445 

69466 

cTsT 



73i35 
73116 
73096 
73076 
73o56 
73o36 
73016 
72996 
72976 
72957 
72937 
72017 
72897 
72877 
72857 
72837 
72817 
72797 
72777 
72757 
72737 
72717 
72697 
72677 
72657 
72637 
72617 
72597 
72577 
72557 
72537 

72517 
72497 
72477 
72457 
72437 
72417 
72397 
72377 
72357 
72337 
72317 
72297 
72277 
72257 
72236 

72216 
72196 
72176 
72i56 
72i36 
72116 
72095 
12075 
72055 
72035 
72015 
71095 
71974 
71954 
71934 
S. 



44 Deff. 



S. 



C. 8. 



69466 
69487 
69508 
69529 
69549 
69570 
69591 
69612 
69633 
69654 
69675 
69696 
69717 

69737 
69758 

69779 
69800 
69821 
69842 
69862 
69883 
69904 
69925 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 

70112 
70132 
70i53 
70174 
70195 
70215 
70236 
70257 
70277 
70298 
70319 
70339 
7o36o 
7o38i 
70401 
70422 
70443 
70463 
70484 
7o5o5 
7o535 
70546 
70567 
70587 
70608 
70628 
70649 



934 

Q14 
894 
873 
853 
833 
8i3 

792 
772 
75? 
732 
711 
691 
671 
65o 
63o 

610 

590 
569 
549 

52i 

5o; 

488 
468 
447 
427 

407 
386 
366 
345 
325 

3o5 
284 
264 
243 

233 
203' 

182 
162 
141 
121 
100 
080 
009 
039 
019 

70998 
70978 
70957 

7^7 
70916 
70896 
70875 
70855 
70834 
70813 
70793 
70772 



to 



46 Deg. 



70670; 70732 
70690; 70731 
70711! 7071 1 

cTsTPsT' 

45 Peg. 



Si 

55' 
54; 
53 
52! 
5il 

4'! 
48,' 

SI 

45 

44 
43 
42 
41 i 

40 1 

3^1 

37 
36' 
35 i 
34! 

33] 
3a i 
3ii 
3ol 

2?! 

26 

35! 

24 i 

23 

22 
21 
20 
19 

:? 

i5 

r4 
i3 

II ! 



72 



TRAVERSE TABLT: 



2 

5' 

3 

a 

« 

-.: « 

1 


iDeg. 


1 

2 


Deg. 


IDeg. 


O 
5' 



3 
n 

? 


liat. 


Dep 
0.00 


Lat. 


Dep. 
0.01 


Lat. 


Dflp 
0.01 


1.00 


1.30 


1.00 


T 


2 


2.00 


0.01 


2.00 


0.02 


2.00 


0.03 


2 


3 


3.00 


0.01 


3.00 


0.03 


3.00 


0.04 


3 


4 


4.00 


0.02 


4.00 


0.03 


4.00 


0.05 


4 


5 


5.00 


0.02 


5.00 1 


0.04 


5.00 


0.07 


5 


6 


6.00 


0.03 


6.00 


0.05 


3.00 


0,08 


6 


7 


7.00 


0.03 


7.00 


0.06 


7.00 


0.09 ' 


7 


8 


8.00 


0.03 


8.00 


0.07 


8.00 


0.10 i 


8 


9 


9.00 


0.04 


9.00 


0.08 


9.00 


0.12 1 


9 


10 


10.00 


0.04 


10.00 


0.09 


10.00 


0.13 1 


10 


11 


11.00 


0.05 


11.00 


0.10 


11.00 


0.14 j 


if 


12 


12.00 


0.05 


12.00 


0.10 


12.00 


0.16 ' 


12 


13 


13.00 


0.06 


13.00 


0.11 


13.00 


0.17 


13 


14 


14.00 


0.06 


14.00 


0.12 


14.00 


0.18 , 


14 


15 ! 


15.00 


0.07 


15.00 


0.13 


15.00 


0.20 


15 


16 


16.00 


0.07 


16.00 


0.14 


16.00 


0.21 


16 


17 1 


17.00 


0.07 


17.00 


0.15 


17.00 


0.22 


17 


18 


18.00 


U 08 


18 !J0 


0.16 


18.00 


0.24 


18 


19 ! 


19.00 


0.08 


19.00 


0.17 


19.00 


0.25 


19 


20. , 


20.00 


0.09 


20.00 


0.17 


20.00 


0.26 


20 


21 1 


21.00 


0.09 


21.00 


0.18 


21.00 


0.27 


21 


22 1 


22.00 


0.10 


22.00 


0.19 


22.00 


0.29 


22 


23 1 


23.00 


0.10 


23.00 


0.20 


23.00 


0.30 


23 


24 


24.00 


0.10 


24.00 


0.21 


24.00 


0.31 


24 


25 i 


25.00 


0.11 


25.00 


0.22 


25.00 


0.33 


25 


26 \ 


26.00 


0.11 


26.00 


0.23 


26.00 


0.34 


26 


27 : 


27.00 


0.12 


27.00 


0.24 


1 27.00 


0.35 


27 


28 1 


28.00 


0.12 


28.00 


0.24 


28.00 


0.37 


28 


29 


29.00 


0.13 


29.00 


0.25 


29.00 


0.38 


29 


80 


30.00 


0.13 


30.00 


0.26 


30.00 


0.39 


30 


31 


31.00 


0.14 


31.00 


0.27 


31,00 


0.41 


31 


32 


32.00 


0.14 


32.00 


0.28 


32.00 


0.42 


32 


33 


33.00 


0.14 


33.00 


0.29 


33.00 


0.43 


33 


34 


34.00 


0.15 


34.00 


0.30 


34.00 


0.45 


34 


35 


35.00 


0.15 


35.00 


0.31 


35.00 


0.46 


35 


36 


36.00 


0.16 


36.00 


0.31 


36.00 


0.47 


36 


37 


37.00 


0.16 


37.00 


0.32 


37.00 


0.48 


37 


38 


38.00 


0.17 


38.00 


0.33 


38.00 


0.50 


38 


39 


39.00 


0.17 


39.00 


0.34 


39.00 


0.51 


39 


40 


40.00 


0.17 


40.00 


0.35 


40.00 


0.52 


40 


41 


41.00 


0.18 


41.00 


0.36 


41.00 


0.54 


■ 41 


43 


42.00 


0.18 


42.00 


0.37 


: 42.00 


0.5£ 


42 


! 43 


43.00 


0.19 


43.00 


0.38 


; 43.00 


0.56 


, 43 


44 


44.00 


0.19 


44.00 


i 0.38 


44.00 


0.53 


44 


45 


45.00 


0.20 


45.00 


1 0.39 


1 45.00 


0.59 


45 


46 


46.00 


0.20 


46.00 


0.4C 


i 46.00 


0.60 


46 


47 


47.00 


0.21 


47.00 


0.41 


: 47.00 


0.62 


47 


4fi 


48.00 


0.21 


48.00 


0.42 


48.00 


0.63 


48 


49 


49.00 


0.21 


49.00 


i 0.43 


49.00 


0.64 


49 


50 


50.00 


0.22 


50.00 


! 0.44 


1 50.00 


0.65 


60 


Distanco.l 


Dep. 


Lat. 


Dep. 


1 Lat. 


i! ^^P- 


Lat. 


6 

c 


891 Deg. 


89J 

t 


Deg. 


n 

m 


Deg. 


5 

CO 

1 



TRAVfeRSE tABLfi 



73 



t3 

do' 

IT*- 

e» 

3 

a 
? 

51 


iDeg. 


1 

3 


Deg. 


1 Deg. 


1 

51 


Lat. 


Dep. 
0.22 


Lat. 


Dep. 
0.45 


Lat. 


1 Dop. 
0.67 


51.00 


51.00 


1 51.00 


52 


52.00 


1 0.23 


62.00 


0.45 


52.00 


0.68 


5% 


53 


53.00 


0.23 


53.00 


0.46 


53.00 


0.69 


i 53 


54 


54.00 


0.24 


54.00 


0.4' 


i 54.00 


' 0.71 


: 64 


55 


55.00 


0.24 


55.00 


0.4o 


65.00 


0.72 


i 65 


56 


56.00 


0.24 


58.00 


0.49 


66.00 


0.73 


' 56 


57 


57.00 


0.25 


67.00 


0.60 


67.00 


0.75 


: 57 


58 


58.00 


0.25 


58.00 


' 0.51 


57.99 


0.76 


58 


59 


59. OC 


0.26 


59.00 


0.61 


' 56.99 


0.77 


, 59 


60 


60. 0( 


0.26 


6C.0C 


0.62 


59.99 


0.79 


60 


61 


61.00 


0.27 


6i.00 


0.53 


60.99 


0.80 


61 


62 


62.00 


0.27 


62.00 


0.54 


61.99 


0.81 


62 


63 


63.00 


0.27 


63.00 


0.55 


62.99 


0.82 


63 


64 


64.00 


0.28 


64.00 


0.56 


63.99 


0.84 


64 , 


65 


65.00 


0.28 


65.00 


0.57 


64.99 


0.85 


65 


66 


66.00 


0.29 


66.00 


0.58 


65.99 


0.86 


66 


67 


67.00 


0.29 


67.00 


0.58 


66.99 


0.88 


67 


68 


68.00 


0.30 


68.00 


0..59 


67.99 


0.S9 


68 


69 


69.00 


0.30 


69.00 


0.60 


68.99 


0.90 


69 


70 


70.00 


0.31 


70.00 


0.61 


69.99 


0.92 


70 


71 


71.00 


0.31 


71.00 


0.62 


70.99 


0.93 


•»i 


72 


72.00 


0.31 


72.00 


0.63 


71.99 


0.94 


72 


73 


73.00 


0.32 


73.00 


0.64 


72.99 


0.96 


73 


74 


74.00 


0.32 


74.00 


0.65 


73.99 


0.97 


74 


75 


75.00 


0.33 


75.00 


0.65 


74.99 


0.98 


75 


76 


76.00 


0.33 


76.00 


0.66 


75.99 


0.99 


76 


77 


77.00 


0.34 


77.00 


0.67 


76.99 


1.01 


77 


78 


78.00 


0.34 


78.00 


0.68 


77.99 


1.02 


78 


79 


79.00 


0.34 


79.00 


0.69 


78.99 


1.03 


79 


80 


80.00 


0.35 


80.00 


0.70 


79.99 


1.05 


80 


81 


81.00 


0.35 


81.00 


0.71 


80.99 


1.06 


"81 


82 


82.00 


0.36 


82.00 


0.72 


81.99 


1.07 


82 


83 


83.00 


0.36 


83.00 


0.72 


82.99 


1.09 


83 


84 


84.00 


0.37 


84.00 


0.73 


83.99 


1.10 


84 


85 


85.00 


0.37 


85.00 


0.74 


84.99 


1.11 


85 


86 


86.00 


0.38 


86.00 


0.75 


86.99 


1.13 


86 


87 


87.00 


0.38 


87.00 


0.76 


86.99 


1.14 


87 


88 


88.00 


0.38 


88.00 


0.77 


87.99 


1.15 


88 


89 


89.00 


0.39 


89.00 


0.78 


88.99 


1.16 


89 


90 


90.00 


0.39 


90.00 


0.79 


89.99 


1.18 ! 


90 


91 


91.00 


0.40 


91.00 


' 0.79 . 


90.99 


1.19 ' 


91 


92 


92.00 


0.40 


92.00 


0.80 


91.99 


1.20 


9S 


93 


93.00 


0.41 


93.00 


0.81 


92. 99 


1.22 


93 


94 


94.00 


0.41 


94.00 


0.82 


93 99 


1 23 


94 


95 


95.00 


0.41 


95.00 


0.83 


94.99 


1.24 


95 


96 


96.00 


0.42 


96.00 


0.84 


95.99 


1.26 


96 


97 


97.00 


0.42 


97.00 


0.85 


96.99 


1.27 


97 


98 


98.00 


0.43 


98.00 


0.86 


97.99 


1.28 


98 


99 


99.00 


0.43 


99.00 


0.86 


98.99 


1.30 


99 


!00 


100.00 


0.44 


100.00 


0.87 


99.99 


1.31 


£00 


d 

d 
to 


Dep. 


Lat. 


Dep. 


Lat. 


Dop. 


Lat. 


d 
















c. 


89? 1 


Depr. 

1 


89^1 


)eg. 


89U 


)eg. 


P 



74 



TRAVERSE TABLE 



j 

o 
s 

1 


1 Deg. 


UD€g. 


1| Dog. 




1| Deg. 


e 

a 


Lat. 
1.00 1 


Dep. 
0.02 


Lat. 1 

Too" 


Dep. 


Lat. 

~r7oo 


Dop 

0.3 


Lat. 

1 


Dep. 


0.02 


1.00 


0.03 


2 


2.00 


0.03 


2.00 


0.04 


2.00 


f,M 


2.00 


0.06 2 


3 3.00 


0.05 


3.00 


0.07 


3 00 


O.O.- 


n.oo 


09, 3 


4 4.00 


0.07 


4.00 


0.09 


i GO 


0.10 


4.00 


0.12 


4 


5 5.00 


0.09 


5.00 


0.11 


5.00 0:i3i 


5.00 


0.15 


5 


6 6.00 


0.10 


6.00 


0.13 


6.r0 


r»,]6 


6.00 


18 


6 


7 


7.00 


0.12 


7.00 


0.15 


7.00 


^M8 


7.00 


0.21 


7 


8 


8.00 


0.14 


8.00 


0.17 


8.00 


0.21 


8.00 


0.25 


8 


9 


9.00 


0.16 


9.00 


0.20 


9.00 


0.24 


9.00 


0.28 


9 


10 
11 


10.00 


17 
0.19 


10.00 


0.22 


10.00 
11.00 


0.26 


10.00 


0.31 


10 
11 


11.00 


11.00 


0.24 


0.28 


10.99 


0.34 


12 


12.00 


0.21 


12.00 


0.26 


12.00 


0.31 


11.99 


0.37 


12 


13 


13.00 


0.23 


13.00 


0.28 


13.00 


0.34 


12.99 


0.40 


13 


14 


14.00 


0.24 


14.00 


0.31 


14.00 


0.37 


13.99 


0.43 


14 


15 


15.00 


0.26 


15.00 


0.33 


14.99 0.39 1 


14.99 


0.46 


15 


16 


16.00 


0.28 


16.00 


0..35 


15.99 


0.42 


15.99 


0.49 


16 


17 


17.00 


0.30 


17.00 


0.37 


16.99 


0.45 


16.99 


0.52 


17 


18 


18.00 


0.31 


18.00 


0.39 


17.99 


0.47 


17.99 


0.55 


18 


19 


19.00 


0.33 


19.00 


0.41 


18.99 


0.50 


18.99 


0.58 


19 


20 

21 


20.00 


0.35 


20.00 


0.44 
0.46 


19.99 


0.52 


19.99 


0.61 


20 
"21 


21.00 


0.37 


21.00 


20.99 


0.55 


20.99 


0.64 


22 


22.00 


0.38 


21.99 


0.48 


21.99 


0.58 


21.99 


0.67 


22 


23 


23.00 


0.40 


22.99 


0.50 


22.99 


0.60 


22.99 


0.70 


23 


24 


24.00 


0.42 


23.99 


0.52 


23.99 


0.63 


23.99 


0.73 


24 


25 


25.00 


0.44 


24.99 


0.55 


24.99 


0.65 


24.99 


0.76 


25 


26 


20.00 


0.45 


25.99 


0.57 


25.99 


0.68 


25.99 


0.79 


26 


27 


27.00 


0.47 


26.99. 


59 


26.99 


0.71 


26.99 


0.83 


27 


28 


28.00 


0.49 


27.99 


0.61 


27.99 


0.73 


27.99 


0.86 


28 


29 


29.00 


0.51 


28.99 


0.63 


28.99 


0.76 


28.99 


0.89 


29 


30 

31 


30.00 
31.00 


0.52 
0.54 


29.99 
30.99 


0.65 


29.99 


0.79 


29.99 


0.92 


30 
31 


0.68 


30.99 


0.81 


30.99 


0.95 


32 


32.00 


0.56 


31.99 


0.70 


31.99 


0.84 


31.99 


0.98 


32 


33 


32.99 


0.58 


32.99 


0.72 


32.99 


0.86 


32.98! 1.01 


33 


34 


33.99 


0.59 


33.99 


0.74 


33.99 


0.89 


33.98 1.04 


34 


35 


34.99 


0.61 


34.99 


0.76 


34.99 


0.92 


34.98 1.07 


35 


36 


35.99 


0.63 


35.99 


0.79 


35.99 


0.94 


35.98 1.10 


36 


37 


36.99 


0.65 


36.99 


0.81 


36.99 


0.97 


36.98 1.13 


37 


38 


37.99 


0.66 


37.99 


0.83 


37.99 


0.99 


37.98 1.16 


38 


39 


38.99 


0.68 


38.99 


0.85 


38.99 


1.02 


38.98 


1.19 


39 


40 


39.99 
40.99 


0.70 
0.72 


39.99 


0.87 


39.99 
40.99 


1.05 
" 1.07' 


39.98 


].22 


40 
'41 


1 41 


40.99' 


0.89 


40.98 


1 25 


1 42 


41.99 


0.73 


41.99 


0.92 


41.99 


1.10 


41.98 


1 28 


42 


43 


42.99 


0.75 


42.99 


0.94 


42.99 


1.13 


42.98 


1.31 43 


44 43.99 


0.77 


43.99 


0.96 


43.99 


1.15 


43.98 


1.34 44 


45 1 44.99 


0.79 


44.99 


0.98 


44.99 


1.18 


44.98 


1.37 45 


46 1 45.99 


0.80 


45.99 


1.00 


45.99 


1.20 


45.98 


1.40 46 


47 146.99 


0.82 


46.99 


1.03 


46.99 


1.23 


46.98 


1.44 47 


iS 47.99 


0.84 


47.99 


1.05 


47.98 


1.26 


47.98 


1 47 


; 48 


Al ,48.99 


0.86 


48.99 


1.07 


48.98 


1.28 


48.98 


1.50 


49 


50 49.99^ 


0.87 


49.99 


1.09 


49.98 


1.31 
Lat 


149.98 
Dep. 


1.53 

Lat. 


6 

c 

ed 

73 

i 


65 

3 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


89 


Deg. 


88| 


Deg. 


SSI Deg. 

i 


88i Deg. 



TRAVERSE TABLK 



75 



a 

P 

s 
p 

51 


1 Deg. 1 

1 


Lat. 


>eg. 


H Deg. 


i 


L>eg. 




o 
? 

'5J 


Lat. 
50.99 


Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.89 


509" 


r 1.11 


.50.98 


1.34 


~50M 


1.56 


52 


51,99 


0.91 


51.99 


1.13 


51.98 


1.36 


51.98 


1..59 


52 


53 


52 99 


0.92 


52.99 


1.10 


52.98 


1.39 


52.98 


1.62 


53 


64 


53 99 


0.94 


53.99 


1.18 


53.98 


1.41 


53.97 


1.65 


54 


55 


o4 99 


0.96 


54 . 91' 


1.20 


54.98 


1.44 


54.97 


1.68 


55 


56 


55.99 


0.98 


55.99 


1.22 


55.98 


1.47 


55.97 


1.71 


56 


57 


56.99 


0.99 


56.99 


1.24 


56 98 


1.49 


56.97 


1.74 


57 


58 


57.99 


1.01 


57.99 


1.27 


.^7.98 


1.52 


57.97 


..77 


58 


59 


58.99 


1.03 


58.99 


1.29 


58.98 


1.54 


58.97 


1.80 


59 


60 

61 


59.99 


1.05 
1.06 


59.99 


1.31 
1.33 


59.98 
60.93 


1.57 
1.60 


59 . 97 


1.83 
1.86 


60 
61 


60.99 


60.99 


60.97 


62 


61.99 


1.08 


61.99 


1.35 


61.9S 


1.62 


61.97 


1.89 


62 


63 


62.99 


1.10 


62.99 


1.37 


62.98 


1.65 


62.97 


1.92 


63 


64 


63.99 


1.12 


63.98 


1.40 


63.93 


1.68 


63.97 


1,95 


64 


65 


64.99 


1.13 


64.98 


1.42 


64.98 


1.70 


64.97 


1 99 


65 


66 


65.99 


1.15 


65.98 


1.44 


85.98 


1.73 


65.97 


2 02 


66 


67 


66.99 


1.17 


66.98 


1.46 


66.98 


1.75 


66.97 


2.05 


67 


68 


67.99 


1.19 


67.98 


1.48 


67.98 


1.78 


67.97 


2.08 


68 


69 


68.99 


1.20 


68.98 


1.51 


68.98 


1.81 


68 97 


2.11 


69 


70 
71 


69.99 
70.99 


1.22 
1.24 


69.98 


1.53 


69.98 


1.83 


69. 97 


2.14 
2.17 


70 
71 


70.98 


1.55 


70.98 


1.86 


70.<J7 


72 


71.99 


1.26 


71.98 


1.57 


71.98 


1.88 


71.97 


2.20 


72 


73 


72.99 


1.27 


72.98 


1.59 


72.97 


1.91 


72.97 


2.23 


73 


74 


73.99 


1.29 


73.98 


1.61 


73.97 


1.94 


73.97 


2.26 


74 


75 


74.99 


1.31 


74.98 


1.64 


74.97 


1.96 


74.97 


2.29 


75 


76 


75.99 


1.33 


75.98 


1.66 


75.97 


1.99 


75.96 


2.32 


76 


77 


76.99 


1.34 


76.98 


1.68 


76.97 


2.02 


76.96 


2.35 


77 


78 


77.99 


1.36 


77.98 


1.70 


77.97 


2.04 


77.96 


2.38 


78 


79 


78.99 


1..38 


78.98 


1.72 


78.97 


2.07 


78.96 


2.41 


79 


80 
81 


79.99 


1.40 


79.98 
80.98 


1.75 
1.77 


79.07 
80.97 


2.09 
2.12 


79.96 


2.44 


80 
81 


80.99 


1.41 


80.96 


2.47 


82 


81.99 


1.43 


81.98 


1.79 


81.97 


2.15 


81.96 


2.50 


82 


83 


82.99 


1.45 


82.98 


1.81 


82.97 


2.17 


82.96 


2.53 


83 


84 


83.99 


1.47 


83.98 


1.83 


83.97 


2.20 


83.96 


2.57 


84 


85 


84.99 


1.48 


84.98 


1.85 


84.97 


2.23 


84.96 


2.60 


85 


86 


85.99 


1..50 


85.98 


1.88 


85.97 


2.25 


85.96 


2.63 


86 


87 


86.99 


1.52 


86.98 


1.90 


86.97 


2.28 


86.96 


2.66 


87 


88 


87.99 


1.54 


87.98 


1.92 


87.97 


2.30 


87.96 


2.69 


88 


89 


88.99 


1.55 


88.98 


1.94 


88.97 


2.33 


88.96 


2.72 


89 


90 
91 


89.99 


1.57 


89.98 
90.98 


1.96 
1.99 


89.97 


2.36 
2.38 


89.96 


2.75 

2.78 


90 
91 


90.99 


1.59 


90.97 


90.96 


92 


91.99 


1.61 


91 98 


2.01 


91.9" 


2.41 


91.96 


2.81 


92 


93 92.99 


1.62 


92.98 


2.03 


! 92.9- 


2.43 


92.96 


2.84 


93 


94 93.99 


1.64 


93.98 


2.05 


93.97 


2.46 


93.96 


2.87 


94 


95 94.99 


1.66 


94.98 


2.07 


94.97 


2.49 


94.96 


2.90 


95 


96 95 99 


1.68 


95.98 


2.09 


95.97 


2.51 


95.96 


2.94 


96 


97 ,90.99 


1.69 


96.98 


2.12 


96.97 


2.54 


96.95 


2.96 


97 


98 


97.99 


1.71 


97.98 


2.14 


97.97 


2.57 


97.95 


2.9S 


9f 


99 


98.98 


1.73 


98.98 


2.16 


98.97 


2.59 


98.95 


' 3 09 


9: 


100 

c 
2 

CQ 

D 


99.98 
Dep. 


1.75 
Lat. 


99.98 


2.18 
Lat. 


99.97 


2.62 


99.95 
Dep. 


, 3.0£ 

' L»t 

1 


100 

§ 

1 

1 


Dep. 


Dep 


Lftt. 


89 1 


3eg:. 


88! Deg. 


881 


Ttv.g. 


88i 


Deg 



76 



TEAVERSE TABLE. 



P 2 Deg. 

S 1 

OS 


2i Deg. 

1 


2| Deg. 


21 Deg. 



e 

" 1 


g Lat. i Dep. 

• 1 


Lat. 


Dep. ; 


Lat. 


Dep 


Lat. 


Dep 


1 1.00 1 0.03 


1.00 


0.04 


1.00 


0.04 


1.00 


0.05 


8 2.0C 0.07 


2.00 


0.08 


2.00 


0.09 


2.00 


0.10 


2 


S 3.0C' 0.10 


3.00 


0.12 


3.00 


0.13 


3.00 


O.U 


a 


1 4.0C 


0.14 


4.00 


0.16 


4.00 


0.17 


4.00 


0.19 


4 


5 5.0C 


0.17 


5.00 


0.20 


5.00 


0.22 1 


4.99 


0.24 


5 


6i 6.0C 


0.21 


6.00 


0.24 


5.99 


0.26 


5.99 


0.29 


6 


7 i 7.00 


0.24 


6.99 


0.27 


6.99 


0.31 


6.99 


0.34 


7 


8; 7.99 


0.28- 


7.99 


0.31 


7.99 


0.35 


7.99 


0.38 


<5 


9 1 8.99 


0.31 


8.99 


0.35 


8.99 


0.39 


8.99 


0.43 


s 


10 9.99 


0.35 


9.99 


0.39 9.99 


0.44 


S.99 


0.48 


10 

II 


11 10.99 


0.38 


10.99 


0.43! 


10.99 


0.48 


10.99 


0.53 


12 11.99 


42 


11.99 


0.47' 


11.99 


0.52 


11.99 


0.58 


12 


13 12.99 


0.45 


12.99 


0.51 , 


12.99 


0.57 


12.99 


0.62 


13 


14 13.99 


0.49 


13.99 


0.55: 


13.99 


0.61 


13.98 


0.67 


14 


15 14.99 


0.52 


14.99 


0.59 


14.99 


0.65 


14.98 


0.72 


15 


16 15.99 


0.56 


15.99 


0.63, 


15.99 


0.70 


15.98 


0.77 


16 


17 ; 16.99 


0.59 


16.99 


0.67, 


16.98 


0.74 


16.98 


0.82 


17 


18 17.99 


0.63 


17.99 


0.7l! 


17.98 


0.79 


17.98 


0.86 


18 


19 18.99 


0.66 


18.99 


0.75: 


18.98 


0.83 


18.98 


0.91 


19 


20 19.99 

21 20.99 


0.70 
0.73 


19.98 


0.79 


19.98 


0.87 
" 0.92 


19.98 
20.98 


0.96 
1.01 


20 
21 


20.98 


0.82' 


20.98 


22 21.99 


0.77 


21.98 


0.86; 


21.98 


0.96 i 


21.97 


1.06 


22 


23 22.99 


O.bO 


22.98 


0.901 


22.98 


1.00 


22.97 


1.10 


23 


24 23.99 


0.84 


23.98 


0.94 


23.98 


1.05 


23.97 


1.15 


24 


25 24.98 


0.87 


24.98 


0.98 


24.98 


1.09 24.97 


1.20 


25 


26 25.9'^ 


0.91 


25.98 


1.02; 


25.98 


1.13 


25.97 


1.25 


26 


27 26.98 


0.94 


26.98 


1.06 


26.97 


1.18 


26.97 


1.30 


27 


28 27.98 


0.98 


27.98 


1.10 


27.97 


1.22 


1 27.97 


1.34 


28 


29 28.98 


1.01 


28.98 


1.14 


28.97 


1.26 


'28.97 


1.39 


29 


30 29.98 


1.05 


29.98 


1.18 
1.22 


29.97 


1.31 


29.97 


1.44 


30 

31 


31 30.98 


1.08 


30.98 


30.97 


1.35 


30.96 


1.49 


32 31.98 


1.12 


31.98 


1.26 31.97 


1.40 


31.96 


1.54 


32 


33 32.98 


1.15 


32.97 


1.30 


32.97 


1.44 


132.96 


1.58 


33 


34 33.98 


1.19 


33.97 


1.33 


1 33.97 


1.48 


33.96 


1.63 


34 


35 34.98 


1.22 


34.97 


1.37 


34.97 


1.53 


34.96 


1.68 


35 


36 35 . 98 


1.26 


35.97 


1.41 


' 35.97 


1.57 


35.96 


1.73 


36 


37 ,36.98 


1.29 


36.97 


1.45 


: 36.96 


1.61 


i 36.96 


1.78 


37 


38 : 37.98 


1.33 


37.97 


1.49 


37.96 


1.66 


137.96 


1.82 


38 


39 33.98 


1.36 


38.97 


1.53 


38.96 


1.70 


38.96 


1.87 


39 


40 39.98 

41 40.98 


1.40 


39.97 


157 


39.96 


1.75 


39.95 


1.92 


40 
41 


1.43 


40.97 


1 61 


40.96 


1.77 


40.95 


1 97 


42 4] 9: 


1 4- 


41.97 


1.65 


41.96 


1.83 


41.95 


2.02 


42 


43 42.97 


1.5t 


42.97 


1.69 


42.96 


1.88 


. 42 . 95 


2.06 


43 


44 43.97 


5-4 


43.97 


1.73 


43.96 


1.92 


43.95 


2.11 


44 


45 44.97 


1.57 


44.97 


1.77 


44.96 


1.96 


44.95 


2.16 


45 


16 45.97 


1.61 


45.96 


1.81 


45.96 


2.01 


45.95 


2.21 


46 


47 46.97 


1.64 


46.96 


1.85 


46.96 


2.05 


46.95 


2.25 


47 


18 47.97 


1.68 


47.96 


1.88 


47.95 


2.09 


47.95 


2.30 


48 


49 4^.97 


1.71 


48.96 


1.92 


48.95 


2.14 


4S.94 


2.35 


49 


50 49.97 
i Oep 


1.74 

Lat. 


49.96 


1.96 
Lat. 


49.95 


2.18 


49.94 


2.40 


50 

6 
U 

e 

X 


Dep. 


Dep. 


Lat, 


Dep. 


Lat. 


1 1 

C 1 88 Deg 


' 87J 


Desr. 


su 


Deg. 


871 


De? 



TRAVERSE TABLE. 



77 





[ 


i 














c 

£■ 

^ 


' 2 Beg, 


2i Deg. 


H 


Deg. 


21 Deg. 


O 

a' 

P 

a 


a 

s 

51 


Lat. 


Dop. 

1.78 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


50.97 


50.96 


2.00 


50.95 


2.22 


50.94 


2,45 


51 


52 


51.97 


).81 


51.96 


2.04 


51.95 


2.27 


51.94 


2.50 


59 


53 162.97 


1.85 


52.96 


2.08 


62.95 


2.3] 


152.94 


2.54 


53 


54 53.97 


1.88 


53.96 


2.12 


53.95 


2.36 


53.94 


2.59 


54 


55 54.97 


1.92 


54.96 


2.16 


54.95 


2.40 


54.94 


2.64 


55 


56.55.97; 1.95 


55.9c 


2.20 


55.95 


2.44 


55.94 


2 69 


56 


£7 '53.97 


1 99 


56. 9? 


2.24 


56.95 


2.49 


56.93 


2.73 


57 


58 , 57 96 


2.02 


57.96 


2.28 


57.94 


2.53 


57.93 


2.78 


58 


59 


58.96 


2.06 


58.95 


2.32 


58.94 


2.57 


58.93 


2.83 


59 


60 
61 


59.96 


2.09 


59.95 


2.36 


59.94 


2.62 
2.66 


59.93 


2.88 
2.93 


60 
61 


60.96 


2.13 


60,95 


2.39 


60.94 


60.93 


62 


61.96 


2.16 


61.95 


2.43 


61.94 


2.70 


61.93 


2.97 


62 


63 


62.96 


2.20 


62.95 


2.47 


62.94 


2.75 


62.93 


3.02 


63 


64 


63.96 


2.23 


63.95 


2.51 


63.94 


2.79 


63.93 


3.07 


64 


65 


64.96 2.27 


64.95 


2.55 


64.94 


2.84 


64.93 


3.12 


65 


66 


65.96 2.30 


65.95 


2.59 


65.94 


2.88 


65.92 


3.17 


66 


67 


66.96 2.34 


66.95 


2.63 


66.94 


2.92 


66.92 


3.21 


67 


68 


67.96 2.37 


67.95 


2.67 


67.94 


2.97 


67.92 


3.26 


88 


69 


68.96 


2.41 


68.95 


2.71 


68.93 


3.01 


68.92 


3.31 


69 


70 
71 


69.96 


2.44 


69.95 


2.75 


69.93 


3.05 


69.92 
70.92 


3.36 


70 
71 


70.96 


2.48 


70.95 


2.79 


70.93 


3.10 


3.41 


72 


71.96 


2.51 


71.94 


2.83 


71.93 


3.14 


71.92 


3.45 


72 


73 


72.96 


2.55 


72.94 


2.87 


72.93 


3.18 


72.92 


3.. 50 


73 


74 


73.95 


2.58 


73.94 


2.91 


73.93 


3.23 


73.91 


3.55 


74 


75 


74.95 2.62 1 


74.94 


2.94 


74.93 


3.27 


74.91 


3.60 


75 


76 


75.95 


2.65 


75 . 94 


2.98 


75.93 


3.31 


75.91 


3.65 


76 


77 


76.95 


2.69 


76.94 


3.02 


76.93 


3.36 


76.91 


3.70 


77 


78 


77.95 


2.72 


77.94 


3.06 


77.93 


3.40 


77.9] 


3.74 


78 


79 


78.95 


2.76 


78.94 


3.10 


78.92 


3.45 


78.91 


3.79 


79 


80 
81 


79.95 


2.79 


79.94 


3.14 


79.92 


3.49 


79.91 


3.84 


80 
'81 


80.95 


2.83 


80.94 


3.18 


80.92 


3.531 


80.91 


3.89 


82 


81.95 


2.86 


81.94 


3.22 


81.92 


3.58 


81.91 


3.93 


82 


83 


82.95 


2.90 


82.94 


3.26 


82.92 


3.62 


82.90 


3.98 


83 


84 


83.95 


2.93 


83.94 


3.30 


83.92 


3.66 


83.90 


4.03 


84 


85 


84.95 


2.97 


84.93 


3.34 


84.92 


3.71 


84.90 


4.08 


85 


86 


85.95 


3.00 


85.93 


3.38 


85.92 


3.75 


8d 9^ 


4.13 


86 


87 


86.95 


3.04 


86.93 


3.42 


86.92 


3.79 


86.90 


4.17 


87 


88 


87.95 


3.07 


87.93 


3.45 


87.92 


3.84 


87.90 


4.22 


88 


89 


88.95 


3.11 


88.93 


3.49 


88.92 


3.88 


88.90 


4.27 


89 


90 


89.95 


3.14 


89.93 


3.53 


89.91 


3.93 


89.90 


4.32 


90 


91 


90.95 


3.18 


90.93 


3.57 


90.91 


3.97 


90.90 


4.37 


91 


93 


91.94 


3.21 


91.93 


3.61 


91.91 


4.01 


91.89 


4.41 


92 


93 


92.94 


3.25 


92.93 


3.65 


92.91 


4.06 


92.89 


4.46 


93 


94 


93.94 


3.28 


93.93 


3.69 


93.91 


4.10 


93.89 


4.51 


94 


95 


94.94 3.32 1 


94.93 


3.73 


94.91 


4.14 


94.89 


4.56 


95 


96 


95.94 


3 35 


95.93 


3.77 


95.91 


4.19 


95.89 


4.61 


96 


97 


96.94 


3.39 


96.93 


3.81 


96.91 


4.23 


96.89 


4.65 


97 


98 


97.94 


3.42 


97.92 


3.85 


97.91 


4.27 


97.89 


4.7^ 


98 


99 


98.94 


3.46 


98.92 


3.89 


98.91 


4.32 


98.89 


4.75 


99 


100 

s 


99.94 


3.49 
Lac. 


99.92 


3.93 


99.91 


4.36 


99.88 


4.80 


100 


Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


a 
S 

m 




































88 Dog. 


87| 


Deg. 


87^ 


Deg. 


87i 


Deg. 


Q 



78 



TRAVERSE TABLE. 



c 

5 

1 

a 

? 

~1 


3E 

Lat. 


leg. 


3^ Deg 


3^ Deg. 


3} Deg. 


1 

m 


Dep. 
0.05 


Lat. 


Dep. 


Lat. 


Dep. 
0.06 


Lat. 


Dep 


1.00 


1. 00 


0.06 


1.00 


1.00 


G.OA 


2 


2.00 


0.10 


2.00 


0.11 


2.00 


0.12 


2.00 


13 


i 


3 


3.00 


0.16 


3.00 


0.17 


2.99 


0.18 


2.99 


3.20 


3 


4 


3.99 


0.21 


3.99 


0.23 


3.99 


0.24 


3.99 


0.26 


4 


6 


4.99 


0.26 


4.99 


0.28 


4.99 


0.31 


4.99 


0.33 51 


6 


5.99 


0.31 


5.99 


0.34 


5.99 


0.37 


5.99 


0.39 





7 


6.99 


0.37 


6.99 


0.40 


6.99 


0.43 


6.99 


0.46 


7 


8 


7.99 


0.42 


7.99 


0.45 


7.99 


0.49 


7.98 


0.62 


8 


9 


8.99 


0.47 


8.99 


0.51 


8.98 


0.55 


8.98 


0.59 


9 


10 
11 


9.99 
10.98 


0.52 


9.98 


0.57 


9.98 


0.61 


9.98 


0.65 


10 

11 


0.58 


10.98 


0.62 


10.98 


0.67 


10.98 


0.72 


12 


11.98 


0.63 


11.98 


0.68 


11.98 


0.73 


11.97 


0.78 


12 


13 


12.98 


0.68 


12.98 


0.73 


12.98 


0.79 


12.97 


0.85 


13 


14 


13.98 


0.73 


13 A8 


0.79 


13.97 


0.85 


13.97 


0.92 


14 


15 14.98 


0.79 


14.98 


0.85 


14.97 


0.92 


14.97 


0.98 


15 


16 15.98 


0.84 


15.97 


0.91 


15.97 


0.98 


15.97 


1.05 


16 


17 16.98 


0.89 


16.97 


0.96 


16.97 


1.04 


16.96 


1.11 


17 


18 


17.98 


0.94 


17.97 


1.02 


17.97 


1.10 


17.96 


1.18 


18 


19 


18.98 


0.99 


18.97 


1.08 


18.96 


1.16 


18.96 


1.24 


19 


20 
21 


19.97 


1.05 


19.97 


1.13 


19.96 


1.22 


19.96 


1.31 


20 
21 


20.97 


1.10 


20.97 


1.19 


20.96 


1.28 


20.96 


1.37 


22 


21.97 


1.15 


21.96 


1.25 


21.96 


1.34 


21.95 


1.44 


22 


23 


22.97 


1.20 


22.96 


1.30 


22.96 


1.40 


22.95 


1.50 


23 


24 


23.97 


1.26 


23.96 


1.36 


23.96 


1.47 


23.95 


1.57 


24 


25 


24.97 


1.31 


24.96 


1.42 


24.95 


1.53 


24.95 


1.64 


25 


26 


25.96 


1.36 


25.96 


1.47 


'25.95 


1.59 


25.94 


1.70 


26 


27 


26.96 


1.41 


26.96 


1.53 


26.95 


1.65 


26.94 


1.77 


27 


28 


27.96 


1.47 


27.95 


1.59 


27.95 


1.71 


27.94 


1.83 


28 


29 


28.96 


1.52 


28.95 


1.64 


28.95 


1.77 


28.94 


1.90 


29 


30 
31 


29.96 


1.57 


29.95 


1.70 


29.94 


1.83 


29.94 


1.96 


30 
31 


30.96 


1.62 


30.95 


1.76 


30.94 


1.89 


30.03 


2.03 


32 


31.96 


• 1.67 


31.95 


1.81 


31.94 


1.95 


31.93 


2.09 


32 


33 


32.95 


1.73 


32.95 


1.87 


32.94 


2.01 


32.93 


2.16 


33 


34 


33.95 


1.78 


33.95 


1.93 


33.94 


2.08 


33.93 


2.22 


34 


35 


34.95 


1.83 


34.94 


1.98 


34.93 


2.14 


34.92 


2.29 


35 


36 


35.95 


1.88 


35.94 


2.04 


35.93 


2.20 


35.92 


2.35 


36 


37 


36.95 


1.94 


36.94 


2.10 


36.93 


2.26 


36.92 


2.42 


37 


38 


37.95 


1.99 


37.94 


2.15 


37.93 


2.32 


37.92 


2.49 


38 


39 


38.95 


2.04 


38.94 


2.21 


38.93 


2.38 


38.92 


2.55 


39 


40 
41 


39.95 


2.09 


39.94 


2.27 


39.93 


2.44 


39.91 


2.62 


40 
4l 


40 94 


2.15 


40.93 


2.32 


40.92 


2.50 


40.91 


2.68 


42 


41.94 


2.20 


41.93 


2.38 


41.92 


2.56 


41.91 


2.76 


n 


43 


42.94 


2.25 


42.93 


2.44 


42.92 


2.63 


42.91 


2.81' 431 


44 


43.94 


2.30 


43.93 


2.49 


43.92 


2.69 


43.91 


2.88 


44 


45 1 44.94 


2.36 


44.93 


2.55 


44.92 


2.75 


44.90 


2.94 


46 


46 '45.94 


2.41 


45.93 


2.61 


45.91 


2.81 


45.90 


3.01 


46 


47,46.94 
48 ^7.93 


2.46 


46.92 


2.66 


46.91 


2.87 


46.90 


3.07 


47 


2.61 


47.92 


2.72 


47.91 


2.93 


47.90 


3.14 


48 


49 48.93 


2.56 


48.92 


2.78 


48.91 


2.99 


48.90 


3.20 


49 


50 

s 

a 


49.93 


2^62^ 
Lat. 


49.92 


2.83 


49.91 


3.05 


49.89 


3.27 


50 

s 

1 


Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




















S 87 1 


)eg. 


86| 


Deg. 


86^ 


Deg. 


86^ 


Deg. 


.3 

Q 



TRAV£RSB •TABLE. 



79 



5 

s 


SDeg. 


3i Deg. 


ai 


Deg. 
Dep. 


31 Deg. 


r 

• 

51 


Lat 

1 


Dep. 


Lat. 


Dep. 


Lat. 


Lat. 


Dep. 


51 : 60.93 


2.67 


60.92 


2.89 


1 50.90 


311 


50.89 


3.34 


52 51.93 


2.72 


51.92 


2.95 


151.90 


3.17 


51.89 


3.40 


53 


53 52.93 


2.77 


52.91 


3.00 


52.90 


3.24 


62.89 


3.47 


53 


54 53.93 


2.83 


63.91 


3.06 


63.90 


3.30 


53.88 


3.63 


54 


55 54.92 


2.88 


64.91 


3.12 


54.90 


3.36 


54.88 


3.60 


55 


5C 56.92 


2.93 


55 91 


3.17 


55.90 


3.42 


55.88 


3.66 


56 


67 56.92 


2.98 


66.91 


3.23 


56.89 


3.48 


.56.88 


3 73 


57 


58 67.92 


3.04 


57.91 


3.29 


57 89 


3.54 


57.88 


3.79 


58 


69 68.92 


3.09 


68.91 


3.34 


58.89 


3.60 


58.87 


3.86 


59 


60 


59.92 


3.14 
3.19 


69.90 
60.90 


3.40 
3.46 


59.89 


3.66 


69.87 
60.87 


3.92 
3.99 


60 
6] 


61 


60.92 


60.89 


3.72 


62 


61.92 


3.24 


61.90 


3.51 


61.88 


3.79 


61.87 


4.05 


62 


63 


62.91 


3.30 


62.90 


3.67 


62.88 


3.85 


62.87 


4.12 


83 


64 


63.91 


3.35 


63.90 


3.63 


63.88 


3.91 


63.86 


4.19 


64 


65 


64.91 


3.40 


64.90 


3.69 


64.88 


3.97 


64.86 


4.25 


65 


66 


66.91 


3.45 


66.89 


3.74 


65.88 


4.03 


65.86 


4.32 


66 


67 


66.91 


3.51 


66.89 


3.80 


66.88 


4-09 


66.86 


4.38 


67 


(58 


67.91 


3.56 


67.89 


3.86 


67.87 


4.15 


67.85 


4.45 


68 


69 


68.91 


3.61 


68.89 


3.91 


68.87 


4.21 


68.85 


4.51 


69 


70 

71 


69.90 


3.66 


69.89 


3.97 


69.87 


4,27 


69.85 


4.58 


70 

71 


70.90 


3.72 


70.89 


4.03 


70.87 


4.33 


70.85 


4.64 


72 


71.90 


3.77 


71.88 


4.08 


71.87 


4.40 


71.85 


4.71 


72 


73 


72.90 


3.82 


72.88 


4.14 


72.86 


4.46 


72.84 


4.77 


73 


74 


73.90 


3.87 


73.88 


4.20 


73.86 


4.. 52 


73.84 


4.84 


74 


75 


74.90 


3.93 


74.88 


4.25 


74.86 


4.68 


74.84 


4.91 


76 


76 


75.90 


3.98 


75.88 


4.31 


75. 86 


4.64 


75.84 


4.97 


76 


77 


76.89 


4.03 


76.88 


4.37 


76.86 


4.70 


76.84 


5.04 


77 


78 


77.89 


4.08 


77.8*7 


4.42 


77,85 


4.76 


77.83 


5.10 


78 


79 


78.89 


4.13 


78.87 


4.48 


78.85 


4.82 


78.83 


5.17 


79 


80 
81 


79.89 


4.19 
4.24 


79.87 


4.64 


79.85 


4.88 


79.83 


6.23 


80 
81 


80.89 


80.87 


4.. 59 


80.85 


4.94 


80.83 


5.30 


82 


81.89 


4.29 


81.87 


4.65 


81.85 


6.01 


81.82 


5.36 


82 


83 


82.89 


4.34 


82.87 


4.71 


82.86 


6.07 


82.82 


6, .43 


83 


84 


83.88 


4.40 


83.86 


4.76 


83.84 


5.13 


83.82 


6.49 


84 


85 


84.88 


4.45 


84.86 


4.82 


84.84 


6.19 


84.82 


5.56 


85 


86 


86.88 


4.60 


85.86 


4.88 


85.84 


5.25 


85.82 


6.62 


86 


87 


86.88 


4.55 


86.86 


4.93 


86.84 


6.31 


86.81 


6.69 


87 


88 


87.88 


4.61 


87.86 


4.99 


87.84 


5.37 


87 .81 


5.76 


88 


89 


88.88 


4.66 


88.86 


5.05 


88.83 


5.43 


88.81 


5.82 


39 


90 
91 


89.88 


4.71 


89.86 


5.10 


89.83 
90.83 


6.49 


89.81 


5.89 


90 
91 


90.88 


4.76 


90.85 


5.16 


6.66 


90.81 


6.95 


92 


91.87 


4.81 


91.85 


5.22 


91.83 


5.62 


91.80 


6.02 


92 


93 


92.87 


4.87 


92.85 


5.27 


92.83 


5.68 


92.80 


6.08 


93 


94 


93.87 


4.92 


93.85 


6.33 


93.82 


5.74 


93.80 


6.15 


94 


9f, 


94.87 


4.97 


94.85 


5.39 


94.82 


5.80 


94.80 


6.21 


95 


9f 


95.87 


5.02 


95.85 


5.44. 


95.82 


5.86 


95.79 


6.28 


96 


97 


96.87 


5.08 


96.84 


6 :0 


96.82 


6.92 


96.79 


6.34 


97 


98 ' 97.87 


6.13 


97.84 


5..'S6 


97.82 


5.98 


97.79 


6.41 


98 


99 98.86 


6.18 


98.84 


6.61 


98.82 


6.04 


98.79 


6.47 


99 


IOC 

9 
1 

Q 


99.86 


5.23 


99.84 


5.67 


99.81 


6.10 
Lat. 


99.79 


6.54 


100 

1 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Dop. 


Lat. 


87 Deg. 


861 Deg. 


86^] 


Deg. 


86i Deg. 



80 



TBAVERSE TABLE. 



1 


4 Deg. 1 


4i Deg. 


H Deg. 


4i Deg. 


K' 


s 
P 
1 * 


















"1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 

0.08 


Lat. 1 

1 


Dep. 

0.08 


1.00 


0.07 


1.00 


0.07 


1.00 


1.00 


2 2.00 


0.14 


1.99 


0.15 


1.99 


16 1 


1.99' 


0.17 


2 


3 2.99 


0.21 


2.99 


0.22 J 


2.99 


24 


2.99! 


0.25 


3 


; 4 a 99 1 


e.28 


3.99 


0.30 I 


3.99 


31 


3.98, 


0.33 


4 


5 4. 99 ^ 


0.35 


4.99 


0.37 1 


4.98 1 


0.39 1 


4.98 


0.41 


5 


.5.99 


0.42 


5.98 


0.44 ' 


5.98 i 


0.47 


5.98 


0.50 


6 


7 , 


6.98 1 


0.49 


6.98 


0.52 ; 


6.9P 


0.55 i 


6.97: 


0.58 


7 


8 


7.98 


0.56 


7.98 


0.59 


7.98 


0.63 


7.97 


0.66 


ri 


9 


8.98: 


0.63 


8.98 


0.67 


8.97 


0.71 


8.97 


0.75 


fl 


1ft 1 


9.98: 


0.70 
0.77 


9.97 


0.74 


9.97 
10.97 


0.78 


9.97 
10.96 


0.83 


10 
11 


111 


10.97' 


10.97 


0.82 


0.86 


0.91 


12 ' 11.97 


0.84 


11.97 


0.89 


11.96 


0.94 


11.96 


0.99 


13: 


13 1 


12.97 


0.91 


12.96 


0.96 


12.96 


1.02 


12.96 


1.08 


J3: 


14 1 


13.97 


0.98 


13.96 


1.G4 


13.96 


1.10 


13.95 


1.16 


14 1 


15 


14.96 


1.05 


14.96 


1.11 


14.95 


1.18 


14.95 


1.24 


15' 


16! 


15.96 


1.12 


15.96 


1.19 


15.95 


1.26 


15.95 


1.32 


16 


17 


16.96 


1.19 


16.95 


1.26 


16.95 


1.33 


16.94 


1.41 


17 


18 


17.96 


1.26 


17.95 


1.33 


17.94 


1.41 


17.94 


1.49 


18 


19 


18.95 


1.33 


18.95 


1.40 


18.94 


1.49 


18.93 


1.57 


19 


20 ; 

21 


19.95 
20.95 


1.40 
1.46 


19.95 


1.48 
1.56 


19.94 


1.57 


19.93 


1.66 


20 


20.94 


20.94 


1.65 


20.93 


1.74 


21 


22 


21.95 


1.53 


21.94 


1.63 


21.93 


1.73 


21.92 


1.82 


22 


23 ' 


22.94 


1.60 


22.94 


1.70 


22.93 


1.80 


22.92 


1.90 


23 


24, 


23.94 


1.67- 


23.93 


1.78 


23.93 


1.88 


23.92 


1.99 


24 


25' 


24.94 


1.74 


24.93 


1.85 


24.92 


1.96 


24.91 


2.07 


25 1 


26 


25.94 


1.81 


125.93 


1.93 


25.92 


2.04 


25.91 


2.15 


26 i 


27 


26.93 


1.88 


26.93 


2.00, 


26.92 


2.12 


26.91 


2.24 


27 


28 


27.93 


1.95 


27.92 


2.08' 


27.91 


2.20 


27.90 


2.32 


28 


29 


28.93 


2.02 


28.92 


2.15 


28.91 


2.28 


28.90 


2.40 


29 


30 
31 


29.93 


2.09 


29.92 


2.22: 


29.91 


2.35 


29.90 


2.48 
2.57 


30 
31 


30.92 


2.16 


130.91 


2.30 


30.90 


2.43 


130.89 


32 


31.92 


2.23 


131.91 


2.37 


31.90 


2.51 


31.89 


2.65 


^2 


33 


32.92 


2.30 


32.91 


2.45 


32.90 


2.59 


32.89 


2.73 


33 


34 


33.92 


2.37 


33.91 


2.52 


33.90 


2.67 


33.88 


2.82 


34 


35 


34.91 


i 2.44 


34.90 


2.59 


34.89 


2.75 


34.88 


2.90 


35 


36 


35.91 


2.51 


35.90 


2.67 


35.89 


2.82 


35.88 


2.98 


36 


37 


36.91 


2.58 


36.90 


2.74 


36.89 


2.90 


36.87 


3.06 


37 


38 


37.91 


2.65 


37.90 


2.82 


37.88 


2.98 


37.87 


3.15 


38 


39 


38.90 


2.72 


38.89 


2.89 


38.88 


1 3.06 


38.87 


3.23 


39 


40 
41 


39.90 
40.90 


2.79 
2.86 


89.89 
40.89 


2.96 


39.88 


1 3.14 


,39.86 


3.31 


40 
"41 


3.04 


40.87 


! 3.22 


! 40.86 


3.40 


42 41.00 


a. S3 


41.88 


3.11 


41.87 


i 3.30 


^41.86 


3.48 


t2 


43 .42.90 


3.00 


42.88 


3.19 


42.87 


: 3.37 


42.85 


3.56 


id 


44 


43.89 


3.07 


43.88 


3.26 


43.86 


3.45 


43.85 


1 3.64 


44 


4fi 


44.89 


3.14 


44.88 


3.33 


44.86 


3.53 


44.85 


{ 3.73 


4f 


46 


45.89 


3.21 


45.87 


3.41 


45.86 


3.61 


45.84 


3.81 


46 


47 146.89 


3.28 


46.87 


! 3.48 


46.86 


3.69 


46.84 


3.89 


47 


4« 47.88 


3.35 


47.87 


3.56 


47.85 


1 3.77 


47.84 


3.97 


4S 


49 


48.88 


3.42 


48.87 


3.63 


48.85 


3.84 


48.83 


4.06 


4h 


50 

' i 

a, 


49.88 


3 49 


49.86 


3.71 


40.86 


3.92 


49.83 


4.14 


50 


Dep. 


Lbt. 


Dop. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


u 

c 


86 


Deg. 


851 


Deg. 


85| Deg. 

i 


85i 


Deg, 


K 

5 



TRAVERSE TABLE. 



81 



s 

s 

cs 

9 

51 


4 Deg. 


44 Deg. 


4^ Deg. 


4| Deg. 


5l 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


50.88 


3.66 


50.86 


3.78 


50.84 


4.00 


50.82 


4.22 


62 


51.87 


3.63 


51.86 


3.85 


51.84 


4.08 


51.82 


4.31 


62 


63 


52.87 


3.70 


52.85 


3.93 


52.84 


4.16 


52.82 


4.39 


53 


54 


53.87 


3.77 


53.85 


4.00 


53.83 


4.24 


53.81 


4.47 


54 


55 


54.87 


3.84 


54.85 


4.08 


54.83 


4.32 


54.81 


4-55 


55 


56 


55.86 


3.91 


55.85 


4.15 


55.83 


4.39 


65.81 


4.64 


56 


57 


56.86 


3.98 


56.84 


4.22 


56.82 


4.47 


56.80 


4.72 


67 


68 


57.86 


4.05 


57.84 


4.30 


57.82 


4.55 


57 80 


4.80 


58 


69 


58.86 


4.12 


58.84 


4.37 


58.82 


4.63 


58.80 


4.89 


59 


60 
61 


59.85 


4.19 


59.84 


4.45 


59.82 


4.7] 


59.79 


4.97 


60 
61 


60.85 


4.26 


60.83 


4.52 


60.81 


4.79 


60.79 


5.05 


62 


61.85 


4.32 


61.83 


4.59 


61.81 


4.86 


61.79 


5.13 


62 


63 


62.85 


4.39 


62.83 


4.67 


62.81 


4.94 


62.78 


5.22 


63 


64 


63.84 


4.46 


63.82 


4.74 


63.80 


5.02 


63.78 


5.30 


64 


65 


64.84 


4.53 


64.82 


4.82 


64.80 


5.10 


64.78 


5.38 


65 


66 


65.84 


4.60 


65.82 


4.89 


65.80 


5.18 


65.77 


5.47 


66 


67 


66.84 


4.67 


66.82 


4.97 


66.79 


5.26 


66.77 


5.65 


67 


68 


67.83 


4.74 


67.81 


5.04 


67.79 


5.34 


67.77 


5.63 


68 


69 


68.83 


4.81 


68.81 


5.11 


68.79 


5.41 


68.76 


5.71 


69 


70 
71 


69.83 
70.83 


4.88 


69.81 


5.19 


69.78 


5.49 


69.76 


5.80 


70 

71 


4.95 


70.80 


5.26 


70.78 


5.57 


70.76 


5.88 


72 


71.82 


5.02 


71.80 


5.34 


71.78 


5.65 


71.75 


5.96 


72 


73 


72.82 


5.09 


72.80 


5.41 


72.77 


5.73 


72.75 


6.04 


73 


74 


73.82 


5.16 


73.80 


5.48 


73.77 


5.81 


73.75 


6.13 


74 


75 


74.82 


5.23 


74.79 


5.56 


74.77 


5.88 


74.74 


6.21 


75 


76 


75.81 


5.30 


75.79 


5.63 


75.77 


5.96 


75.74 


6.29 


76 


77 


76.81 


5.37 


76.79 


5.71 


76.76 


6.04 


76.74 


6.38 


77 


78 


77.81 


5.44 


77.79 


6.78 


77.76 


6.12 


77.73 


6.46 


78 


79 


78.81 


5.51 


78.78 


5.85 


78.76 


6.20 


78.73 


6.54 


79 


80 
81 


79.81 


5.58 


79.78 


5.93 


79.75 

80.75 


6.28 
6.36 


79.73 


6.62 


80 
81 


80.80 


5.65 


80.78 


6.00 


80 . 72 


6.71 


82 


81.80 


5.72 


81.78 


6.08 


181.75 


0.43 


81.72 


6.79 


82 


83 


82.80 


5.79 


82.77 


6.15 


82.74 


6.51 


82.71 


6.87 


83 


84 


83.80 


5.86 


83.77 


6.23 


83.74 


6.59 


83.71 


6.96 


84 


85 


84.79 


5.93 


84.77 


6.30 


84.74 


6.67 


84.71 


7.04 


85 


86 


85.79 


6.00 


85.76 


6.37 


85 . 73 


6.75 


85.70 


7.12 


86 


87 


86.79 


6.07 


86.76 


6.45 


86.73 


6.83 


86.70 


7.20 


87 


88 


87.79 


6.14 


87.76 


6.52 


87.73 


6.90 


87.70 


7.29 


88 


89 


88.78 


6.21 


88.76 


6.60 


88.73 


6.98 


88.70 


7.37 


89 


90 
91 


89.78 


6.28 


89.75 
90.75 


6.67 


89.72 


7.06 
7.14 


89.69 


7.45 


90 


90.78 


6.35 


6.74 


90.72 


90.69 


7.54 


91 


92 


91.78 


6.42 


91.75 


6.82 


91.72 


7.22 


91.68 


7.62 


92 


93 


92.77 


6.49 


92.74 


6.89 


92.71 


7.30 


92.68 


7.70 


y? 


H 


93.77 


6.56 


93.74 


6.97 


93.71 


7.38 


93.68 


7.78 


9h 


'^5 


94.77 


6.63 


94.74 


7.04 


94.71 


7.45 


94.67 


7.87 


95 


96 '95.77 


6.70 


95.74 


7.11 


95.70 


7.53 


95.67 


7.95 


96 


f)7 


96.76 


6.77 


96.73 


7.19 


96.70 


7.61 


96.67 


8.03 


97 


98 


97.76 


6.84 


97.73 


7.26 


97.70 


7,69 


97.66 


8.12 


38 


99 


98.76 


6.91 


98.73 


7.34 


98.69 


7.77 


98.66 


8.20 


99 


100 

? 

a 

1 


99.76 


6.98 


99.73 


7.41 


99.69 


7.85 
Lat. 


99.66 


8.28 


100 

i 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 
















Q 86 1 


)eg. 

i 


851 


Deg. 


85| Deg. 


85i 


Deg. 


s 



82 



TBAVERSB TABLE. 



s 

g 

O 
O 


5 Deg. 


5i Deg. 


5j Deg. 


5J Deg. 1 

i 




5* 


Lat. 
1.00 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


3 



0.09 


1.00 


0.09 


1.00 


0.10 


0.99 


O.IC 1| 


2 


1.99 


0.17 


1.99 


0.18 


1.99 


0.19 


1.99 


0.20 21 


3 2.99 { 


0.26 


2.99 


0.27 


2.99 


0.29 


2.98 1 


0.30 1 


3' 


4 3.98 


0.35 


3.98 


0.37 


3.98 


0.38 


3.98. 


0.40! 


4 


5 4.98 


0.4A 4.98 


0.46 


4.98 


0.48 


4.97 


0.60 1 


5 


6i 5.98 


0.521 


5.97 


0.55 


5.971 


0.58 


5.97 


0.60 


6 


7 6.97 


0.61! 


0.97 


0.64 


6.97 


0.67 


6.96 


0.70 


7 


8' 7.97 


0.70 1 


7.97 


0.73 


7.96 


0.76 


7.96 


0.80 


8 


9 8.97 


0.78 1 


8.96 


0.82 


8 96 


0.86 


8.95 


0.90 


9 


10 
11 


9.96 
10.96 


0.87i 
0.96! 


9.96 


0.92 


9.95 
10.95 


0.96 ; 


9.95 


1.00 


10 


10.95 


1.01 


1.05 


10.94 


1.10 


11 


12 


I1.95 


1.05 


11.93 


1.10 


11.94 


1.15 


11.94 


1.20 


12 


13 


12.95 


1.13^ 


12.95 


1.19 


12.94 


1.25 


12.93 


1.30 


13 


14 


13.95 


1.22 


13.94 


1.28 


13.94 


1.34 


13.93 


1.40 


14 


15 


14.94 


1.31 


14.94 


1.37 


14.93 


1.44 


14.92 


1.50 


15; 


16 


15.94 


1.39 


15.93 


1.46 


15.93 


1.53 


15.92 


1.60 


1« 


17 


16.94 


1.48 


16.93 


1.56 


16.92 


1.63 


16.91 


1.70 


17 


18 


17.93 


1.57 


17.92 


1.65 


17.92 


1.73 


17.91 


1.80 


18 


19 


18.93 


1.66 


18.92 


1.74 


18.91 


1.82 


18.90 1.90 


19 


20 
21 


19.92 


1.74 


19.92 


1.83 


19.91 


1.92 


19.90 


2.00 1 


20 

21 


20.92 


1.83 


20.91 


1.92 


20.90 


2.01 


20.89 


2.10 


22 1 21.92 


1.92 


21.91 


2.01 


21.90 


2.11 


21.89 


2.20 


22 


23 22.91 


2.00 


22.90 


2.10 


22.89 


2.20 


22.88 


2.30 


23 


24 


23.91 


2.09 


23.90 


2.20 


23.89 


2.30 


23.88 


2.40 


24 


25 


24.90 


2.18 


24.90 


2.29 


24.88 


2.40 


24.87 


2.50 


25 


26 


25.90 


2.27 


25.89 


2.38 


25.88 


2.49 


25.87 


2.60 


26 


27 


26.90 


2.35 


26.89 


2.47 


26.88 


2.59 


26.86 


2.71 


27 


28 


27.89 


2.44 


127.88 


2.56 j 


27.87 


2.88 1 


27.86 


2.81 


28 


29 


28.89 


2.53 


28.88 


2.65 1 


28.87 


2.78 1 


28.85 


2.91 


29 


30 


29.89 


2.61 


29.87 


2.75 


29.86 


2.88 ! 


29.85 
30.84 


3.01 


3i) 

1 


31 


30.88 


2.70 


30.87 


2.84 


30.86 


2.97 


3.11 


32 


31.88 


2.79 


31.87 


2.93 


31.85 


3.07 


31.84 


3.21 


32 


33 


32.87 


2.88 


32.86 


3.02 


1 32.85 


3.16 


32.83 


3.31 


33 


34 


33.87 


2.96 


33.86 


1 3.11 


33.84 


3.26 


33.83 


3.41 


34 


35 


34.87 


3.05 


'34.85 


i 3.20 


34.84 


3.35 


34.82 


3.51 


35 


36 


35.86 


3.14 


! 35.85 


1 3.29 


35.83 


3.45 


35.82 


3.61 


36 


37 


36.86 


3.22 


i 36.84 


! 3.39 


! 36.83 


3.55 


36.81 


3.71 


37 


38 


37.86 


3.31 


37.84 


1 3.48 


37.83 


3.64 


37.81 


3,81 


38 


39 


38.85 


3.40 


i 38.84 


3.57 


38.82 


3.74 


38.80 


3.91 .39 


40 

41 


39.85 
40.84 


3.49 
3.57 


39.83 


3.66 


39.82 


3.83 


39.80 1 4.01 


40 


40.83 


i 3.75 


40.81 


3.93 


40.79 j 4.11 


41 


* 42 


41.84 


3.66 


41.82 


3.84 


41.81 


4.03 


41.79 ! 4.21 ! 42 


43 


42 84 


3.75 


42 . 82 


i 3.93 


42.80 


4.12 


42.78 


4.31 43 


44 


43.83 


1 3.83 


43.82 


i 4.03 


43.80 


4.22 


43.78 


4,41 1 44 


45 


44.83 


3.92 


44.81 


! 4.12 


44.79 


4.31 


44.77 


4 51 


1 45 


46 


45.82 


4.01 


45.81 


4.21 


45.79 


4.41 


45.77 


4 61 


46 


17 


46.82 


4.10 


46.80 


4.30 


46.78 


4.50 


46.76! 4.71 


47 


48 


47.82 


4.18 


47.80 


4.39 


47.78 


4.60 


47.76! 4.81 


48 


4Q 48.81 


; 4.27 


48,79 


1 4.48 


48.77 


4.70 


48.75, 4.91 


49 


60 

i 

93 

s 


49.81 
Dep. 


1 4.36 
Lat. 


49.79 


1 4.58 


49.77 


4.79 
Lat. 


49.75 


5.01 


50 


Dep. 


1 

1 Lat 


Dep. 


Dep. 


1 Lat. 

1 


i 9 

: 5 


85 


Deg. 


841 


Deg. 


84^ 


Deg. 


i 


Deg. 



TRA\£RSB TAitLfi. 



83 



o 


5Deg. 


6i Deg. 


H 


Deg. 


3|Deg 


C 


P 

s 














1 

6: 


Lat 


Dop. 


Lat. 


Dep. 


Lat. 
50.77 


Dep. 


Lat. 


Dep. 


50.81 


4.44 


60.79 


4.67 


4.89 


50 74 


6.11 


fi2 


51.80 


4.53 


51.78 


4.76 


151.76 


4.98 


61 74 


6.21 55!l 


fi3 


52.80 


4.62 


52.78 


4.85 


52.76 


5.08 


62 73! 5.31 


6a 


fi4 


53.79 


4.71 


53.77 


4.94 


53.75 


5.18 


53 73 


5.41 


54 


5ft 


54.79 


4.79 


54.77 


5.03 


54.75 


5.27 


54.72 


5.51 


55 


66 


55.79 


4 88 


55.77 


5.12 


55.74 


5.. 37 


55.72 


5.61 


56 


67 


56.78 


4.97 


56.76 


6.22 


56.74 


5.46 


56.71 


5.71 


57 


58 157.78 


5.06 


57.76 


5.31 


57.73 


5.56 


67.71 


5.81 58 


59 


58.78 


5.14 


58.75 


5.40 


58.73 


6.65 


58.70 


5.91 59 


GO 
fil 


59.77 
60.77 


5.23 


59.75 


6.49 


59.72 


5.76 


69.70 


6.01 


60 


5.32 


60.74 


5.58 


60.72 


5.85 


60.69 


6.11 


61 


62 


61.76 


5.40 


61.74 


5.67 


61.71 


5.94 


61.69 


6.21 


62 


63 


62.76 


5.49 


62.74 


5.76 


62.71 


6.04 


62.68 


6.31 


63 


64 


63.76 


5.68 


63.73 


5.86 


63.71 


6.13 


63.68 


6.41 


64 


65 


64.75 


5.67 


64.73 


5.95 


64.70 


6.23 


64.67 


6.51 


65 


66 


65.75 


5.76 


65.72 


6.04 


65.70 


6.33 


65.67 


8.61 


66 


67 


66.75 


5.84 


66.72 


6.13 


66.69 


6.42 


66.66 


6.71 


6? 


68 


67.74 


5.93 


67.71 


6.22 


67.69 


6.52 


67.66 


6.81 


68 


69 


68.74 


6.01 


68.71 


6.31 


68.68 


6.61 


68.65 


6.91 


69 


70 

71 


69.73 


6.10 


69.71 


6.41 


69.68 


6.71 


69.65 


7,01 


70 
71 


70.73 


6.19 


70.70 


6.50 


70.67 


6.81 


70.64 


7.11 


72 


71.73 


6.28 


71.70 


6.59 


71.67 


6.90 


71.64 


7.21 


72 


73 


72.72 


6.36 


72.69 


6.68 


72.66 


7.00 


72.63 


7.31 


73 


74 


73.72 


6.46 


73.69 


6.77 


73.66 


7.09 


73.63 


7.41 


74 


75 


74.71 


6.54 


74.69 


6.86 


74.65 


7.19 


74.62 


7.51 


75 


76 


75.71 


6.62 


75.68 


6.96 


75.66 


7.28 


76.62 


7.61 


76 


77 


76.71 


6.71 


76.68 


7.06 


76.65 


7.38 


76.61 


7.71 


77 


78 


77.70 


6.80 


77.67 


7.14 


77.64 


7.48 


77.61 


7.81 


78 


79 


78.70 


6.89 


78.67 


7.23 


78.64 


7.57 


78.60 


7.91 


79 


80 

81 


79.70 
80.69 


6.97 


79.66 


7.32 


79.65* 


7.67 


79.60 


8.02 


80 
81 


7.06 


80.66 


7.41 


80.63 


7.76 


80.59 


8.12 


82 


81.69 


7.15 


81.66 


7.60 


81.62 


7.86 


81.59 


8.22 


82 


83 


82.68 


7.23 


82.65 


7.59 


82.62 


7.96 


82.58 


8.32 


83 


84 


83.68 


7.32 


83.65 


7.69 


83.61 


8.05 


83.58 


8.42 


84 


85 


84.68 


7.41 


84.64 


7.78 


84.61 


8.16 


84.67 


8.52 


85 


86 


85.67 


7.50 


86.64 


7.87 


85.60 


8.24 


85.57 


8.62 


86 


87 


86.67 


7.58 


86.64 


7.96 


86.60 


8.34 


86.. 56 


8.72 


87 


88 


87.67 


7.67 


87.63 


8.05 


87.59 


8.43 


87.56 


8.82 


88 


89 


88.66 


7.76 


88.63 


8.14 


88.59 


8.53 


88.55 


8.92 


89 


90 


89.66 


7.84 


89.62 


8.24 


89.59 


8.63 


89.55 


9.02 


90 
91 


90.65 


7.93 


90.62 


8.33 


90.58 


8.72 


90.54 


9.12 


92 


91 .65 


8.02 


91.61 


8.42 


91 .58 


8.82 


91.54 


9.22 


92 


93 


92.65 


8.11 


92.61 


8.51 


92.57 


8.91 


92.63 


9.32 


93 


94 


93.64 


8.19 


93.61 


8.60 


93.57 


9.01 


93.63 


9.42 


94 


95 


94.64 


8.28 


94.60 


8.69 


94.56 


9.11 


94.62 


9.52 


95 


96 


95.63 


8.37 


95.60 


8.78 


96.56 


9.20 


95.52 


9.62 96 


97 


96.63 


8.46 


96.59 


8.88 


96.55 


9.30 


96.51 


9.72 97 


98 


97 63 


8.64 


97.59 


8.97 


97.55 


9.39 


97.51 


9.82 1 98 


99 


98 52 


6 63 


98.59 


9.06 


98.64 


9.49 


98.50 


9.92 99 


i30 

1 

■ 


99.62 


8.72 


99.58 


9.15 


99.54 


9.58 


99.50 


10.02 


100 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat 


1 

1 


85Deg. 


84} Deg. 


84^ Deg. 

i 


84i 


Deg 



•^0 



84 



TRAVERSE TA-BLE. 



i 

o 
s. 

g 1 


6 Deg. 


6^ Deg. 


1 
^ Deg. 1 


6| Deg. 


m 

P 



1 


Lat. 


Dep. 


Lat. 1 

1 


1 
Dop. j 


Lat. 

0.99 


Dep. 


Lat. 


Dop. 


0.99 


0.10 


0.99 i 


0.11 


0.11 


0.99 


0.12 


% 


1 99 


0.21 


1.99 


0.22 


1.99 


0.23 


1.99 


24 


% 


3 


3 98 


0.31 j 


2.98 


0.33 


2.98 


0.34 


2.98 1 35 


3 


4 


3 98 


0.411 


3.98 ! 


44 


3.97 


0.45 


3.97 1 0.47 


4 


5 


4.97 


0.52 


4.97 


0.54 


4.97 , 


0.57 


4.97' 3.59 


5 


6 


5.97 


0.63 


5.96 


0.65 


5.96 1 


0.68 


5.96 0.71 


8 


7 


6.96 


0.73 


6.96 j 


0.76: 6.96 


0.79 


6.95: 0,82 


/ 


8 


7.96 


0.84 


7.95 i 


0.87 


7.95 


0.91 


7.94! 0.94 


8 


9 


8.95 


0.94 


8.95i 


0.98 


8.94 


1.02 


8.94; 1.00 9 i 


10 

11 


9.95 


1,05 j 


9.94 


1.09; 


9.94 
10.93 


1.13 


9.93 


1.18 -0 1 


10.94 


1.15' 


10.93 


1.20 


1.25 


10.92 


1.29 


11 


12 


11.93 


1.25 


11.93 


1.31 


11.92 


1.36 


11.92 1.41 


12 


13 


12.93 


1.36 


12.92 


1.42 


12.92 


1.47 


12.91 


1.53 


13 


14 


13.92 


1.46 


13.92 


1.52, 


13.91 


1.59 


13.90 


1.65 


14 


15 


14.92 


1.57 


14.91 


1.63 


14.90 


1.70 


14.90 


1.76 


15 


16 


15.91 


1.67 


15.90 


1.74: 


15.90 


1.81 


15.89 1.88 16 i 


17 


16.91 


1.78 


16.90 


1.851 


16.89 


1.92 


16.88 


2 00 17 


18 


17.90 


1.88 


17.89 


1.96! 


17.88 


2.04 


17.88 


2.12 : 18 


19 


18.90 


1.99 


18.89 


2.07' 


18.88 


2.15 


18.87 


2.23 19 


20 
21 


19.89 


2.09 


19.88 


2.18| 


ir.87 


2.26 


19.86 


2.35 


20 


20.88 


2.20 


20.88 


2.29! 


20.87 


2.38 


20.85 


2.47 


21 


22 


21.88 


2.30 


21.87 


2.40 1 


21.86 


2.49 


21.85 


2.59 ' 22 


23 


22.87 


2.40 


22.86 


2.50 i 


22.85 


2.00 


22.84 


2.70 j 23 


24 


23.87 


2.51 


23.86 


2.61 


23.85 


2.72 


23.83 


2.82 24 


25 


24.86 


2.61 


24.85 


2.72 


24.84 


2.83 


24.83 


2.94 25 


26 


25.86 


2.72 


25.85 


2.83 


25.83 


2.94 


25.82 


3.06 26 


27 


26.85 


2.82 


26.84 


2.94 


,26.83 


3.06 


26.81 


3.17 j 27 


28 


27.85 


2.93 


27.83 


3.05 


27.82 


3.17 


27.81 


3.29 ' 28 


29 


28.84 


3.03 


28.83 


3.16 


28. SI 


3.28 


28.80 


3.41 ' 29 


30 
31 


29.84 


3.14 


29.82 


3.27 


29.81 


3.40 


29.79 


3.53 30 


30.83 


3.24 


30.82 


3.37 


30.80 


3.51 


30.79 


3.64 i 31 


32 


31 82 


3.34 


31.81 


3.48 


31.79 


3.62 


:31.78 


3.76! 32 


33 


32.82 


8.45 


32.80 


3.59 


32 . 79 


3.74 


32.77 


3.88 ! 33 


34 


33.81 


3.55 


33 30 


3.70 


133.78 


3.85 


133.76 


4.00 ' 34 


35 


34.81 


3.66 


34.79 


3.81 


' 34.78 


3.96 


34.76 


4.11 ! 35 


36 


35.80 


3.76 


35.79 


3.92 


35.77 


4.08 


35.75 


4.23 1 36 


37 


36.80 


3.87 


36.78 


4.03 


36.76 


4.19 


36.75 


4.35 I 37 


38 


37.79 


3.97 


37.77 


4.14 


37.76 


4.30 


37.74 


4.47 i 38 


39 


38.79 


4.08 


38.77 


4.25 


i 38 . 75 


4.41 


38.73 


4,58 ] 39 


40 
41 


39.78 


4.18 
4.29 


39.76 


4.35 


39.74 
: 40 . 74 


4.53 

4.64 


39.72 


4-70 ' 40 
4 82 !'4f 


40.78 


40.76 


4 46 


40.72 


42 


41.77 


■ 4.39 


41.75 


4.57 


.41.73 


4.76 


41.71 j 4.941 42 


43 


42.76 


; 4.19 


42.74 


4.68 


; 42.72 


4.87 


42.701 5.05 1 43 


44 


43.76 


1 4.60 


43.74 


4.79 


j 43.72 


4.98 


43.70 5.17 ; 44 


45 


44.75 


; 4.70 


44.73 


4.90 


'44.71 


5.09 


44.69! 5.29 : 4.5 


46,46.75 


' 4.81 


45.73 


5.01 


; 45.70 


5.21 


45.68 i 5.4] 46 


47 46.74 


4.91 


46.72 


5.12 


46.70 


5.32 ii46.67 5 52 47 j 


•48 147.74 


5.02 


47.71 


5.83 


47.69 


5.43 II 47.67 5.64 


48 


49 48.73 


6.12 


48.71 


5.34 


48.69 


5.55 


48.06] 5.76 


49 


60 

s 

e 

O 
1 


49.73 


' 5.23 

! Lat. 


49.70 


5.44 


49.68 


5.66 


49.65 


5.88 


50 

I'i 

a 

s 

5 


Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat, 


84 

■ ■ ■ .. 


D«g. 


831 


Deg. 


83J 


Deg. 


83^ Deg. 



TRAVERSE TABLE. 



85 



O 6 Deg. 


H Deg. 


61 Deg 


61 Deg. 


O 

s 


1 












O 
? 

"51 


Lat 


Dep. 


Lat. 


Dep. 


J at. 


Dep. 


Lat. 


Dep. 


5 60.72 


5.33 


60.70 


5.65 


60.67; 


5.77 


50.66 


5,99 


55 51.72 


5.44 


51.69 


5.66 


51.67i 


5.89 


51.64 


6.11 


612 


53 52.71 


5.54 


.52.68 


5.77 


52.66 


6-00 


52.63 


6.2.^ 


53 


64 53.70 


5.64 


53.68 


5.88 


53.60 


6.11 


53.63 


6.35 


54 


55 


54.70 


6.75 


54.67 


5.99 


54.65 


6.23 


54.62 


6.46 


65 


56 


55.69 


5.85 


55.67 


6.10 


55.64 


6 34 


55.61 


6.68 


56 


57 


56.69 


5.96 


56.66 


6.21 


56.63 


e-is 


56.60 


6.70 


67 


58 


57.68 


6.06 


57.66 


6.31 


57.63 


6.57 


57.60 


6.82 


68 


59 


58.68 


6.17 


58.65 


6.42 


58.62 


5.68 


58.59 


6.93 


69 


60 
*61 


59.67 


6.27 


59.64 


6.53 


59.61 


6.79 


59.58 


7.05 


60 

61 


60.67 


6.38 


60.64 


6.64 


60.61 


6.91 


60.58 


7.17 


62 


61.66 


6.48 


61.63 


6.75 


61.60 


7.02 


61.57 


7.29 


62 


63 


62.65 


6.59 


62.63 


6.86 


02.60 


7.13 


62.56 


7.40 


63 


64 


63.65 


6.69 


63.62 


6,97 


63.59 


7.25 


63.56 


7.52 


64 


65 


64.64 


6.79 


64.61 


7.08 


64.58 


7.36 


64.55 


7.64 


65 


66 


65.64 


6.90 


65.61 


7 19 


65.58 


7.47 


65.64 


7.76 


66 


67 


66.63 


7.00 


66.60 


7.29 


66.57 


7 58 


66.54 


7.88 


67 


68 


67.63 


7.11 


67.60 


7.40 


67.56 


7.70 


67.53 


7.99 


68 


69 


68.62 


7.21 


68.59 


7.51 


68.56 


7.81 


68.52 


8.11 


69 


70 
71 


69.62 


7.32 


69.58 


7.62 


69.55 


7.92 


69.51 


8.23 


70 

71 


70.61 


7.42 


70.58 


7.73 


70.54 


8.04 


70.51 


8.35 


72 


71.61 


7.63 


71.57 


7.84 


71.54 


8.15 


71.60 


8.46 


72 


73 


72.60 


7.63 


72.57 


7.95 


72.53 


8.26 


72.49 


8.58 


73 


74 


73.59 


7.74 


73.56 


8.06 


73.52 


8.38 


73.49 


8.70 


74 


75 


74.59 


7.84 


74.55 


8.17 


74.52 


8.49 


74.48 


8.82 


75 


76 


75 58 


7.94 


75.55 


8.27 


75.51 


8.60 


76.47 


8.93 


76 


77 


76.68 


8.05 


76.54 


8.38 


76.51 


8.72 


76.47 


9.05 


77 


78 


77.57 


8.15 


77.54 


8.49 


77.60 


8.83 


77.46 


9.17 


78 


79 


78.57 


8.26 


78.53 


8.60 


78.49 


8.94 


78.45 


9.29 


79 


80 
81 


79.56 


8.36 


79.53 


8.71 


79.49 


9.06 


79.45 


9.40 


80 

8} 


80.56 


8.47 


80.62 


8.82 


80.48 


9.17 


80.44 


9.52 


82 


81 55 


8.67 


81.61 


8.93 


81.47 


9.28 


81.43 


9.64 


82 


83 


82 55 


8.68 


82.51 


9.04 


82.47 


9.40 


82.42 


9.76 


83 


84 


83.54 


8.78 


83.50 


9.14 


83-46 


9.51 


83.42 


9.87 


84 


85 


84.53 


8.88 


84.50 


9.25 


84.45 


9.62 


84.41 


9.99 


85 


86 


85.53 


8.99 


85.49 


9.36 


85.45 


9.74 


86.40 


10.11 


86 


87 


86.52 


9.09 


86.48 


9.47 


86.44 


9.85 


86.40 


10.23 


87 


88 


87.52 


9.20 


87.48 


9.68 


87.43 


9.96 


87.39 


10.34 


88 


89 


88.51 


9.30 


88.47 


9.69 


88.43 


10.08 


88.38 


10.46 


89 


90 
91 


89.51 


9.41 


89.47 


9.80 


89.42 


10.19 


89.38 


10.58 


90 


90.50 


9.61 


90.46 


9.91 


90.42 


10.30 


90.37 


10.70 


92 


91.60 


9.62 


91.46 


10.02 


91.41 


10.41 


91.36 


0.81 


92 


93 


92.49 


9.72 


92.45 


10.12 


92.40 


10.53 


92.36 


10.93 


93 


94 


93.49 


9.83 


93.44 


10.23 


93.40 


10.64 


93.35 


11.05 


94 


9b 


94.48 


9.93 


94.44 


10.34 


94.39 


10.75 


94.34 


.1.17 


95 


96 


95.47 


10.03 


95.43 


10.46 


95.38 


10.87 


95.33 


11.28 


96 


97 


96.47 


10.14 


96.42 


10.66 


96.38 


10.98 


96.33 


11.40 


97 


98 


97.46 . 10.24 


97.42 


10.67 


97.37 


11.09 


97.32 


U.52 


98 


99 


98.46 


10.35 


98.41 


10.78 


98.36 


11.21 


98.31 


U.64 


99 


100 

'i 


99.45 
Dep. 


10.45 


99.41 


10.89 


99.36 


11.32 


99.31 


il.75 


100 


Lat. 


Dop. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 




1 














■ 


c 












♦J 


84 Deg. 


831 Deg. 


8^ 


Deg. 


83i Deg. 


Q 



86 



TRAVEBSE TABLE- 



5 
s 

s 

O 

a 

T 


7 Dog. 


7i Deg. 


7iDeg 


71 Deg. 


C 

J 


Lat. 

"0.99" 


Dep. 


Lat. 


D«p. 


Lat. 


Dep. 


Lat. 


Dep. 
"0TT3" 


0.12 


0.99 


0.13 


0.99 


0.13 


0.99 


2 1.99 


0.24 


1.98 


0.25 


1.98 


0.26 


1.98 


0.U7 


8 


3 


2.98 


0.37 


2.98 


0.38 


2.97 


0.39 


2.97 


0.40 


a 


4 


3.97 


0.49 


3.97 


0.50 


3.97 


0.52 


3.96 


0.54 4l 


5 


4 96 


0.61 


4.96 


0.63 


4.96 


0.65 


4.95 


0.67 


6 


6 


5.96 


0.73 


5.95 


0.76 


5.95 


0.78 


5.95 


0.81 


6 


7 


6.95 


0.85 


6.94 


0.88 


6.94 


0.91 


6.94 


0.94 7 


d 


7.94 


0.97 


7.94 


1. 01 


7.93 


1.04 


7.93 


1.08 8 


9 


8.93 


1.10 


8.93 


1.14 


8.92 


1.17 


8.92 


1.21 9 


10 
11 


9.93 
10.92 


1.22 


9.92 


1.26 


9.91 


1.31 


9.91 


1.35 


10 


1.34 


10.91 


1.39 


10.91 


1.44 


10.90 


1.48 


n 


12 


11.91 


1.46 


11.90 


1.51 


11.90 


1.57 


11.89 


1.62 


12 


13 


12.90 


1.58 


12.90 


1.64 


12.89 


1.70 


12.88 


1.75 


13 


14 


13.94) 


1.71 


13.89 


1.77 


13.88 


1.83 


13.87 


1.89 


14 


15 


14.89 


1.83 


14.88 


1.89 


14.87 


1.96 


14.86 


2.02 


15 


16 


15.88 


1.95 


15.87 


2.02 


15.86 


2.09 


15.85 


2.16 


16 


17 


16.87 


2.07 


16.86 


2.15 


16.85 


2.22 


16.84 


2.29 


17 


18 


17.87 


2.191 


17.86 


2.27 


17.85 


2.35 


17.84 


2.43 


18 


19 


18.86 


2.32 1 


18.85 


2.40 


18.84 


2.48 


18.83 


2.56 


19 


20 
21 


19.85 
20.84 


2.44: 


19.84 


2.52 


19.83 


2.61 


19.82 


2.70 


20 
21 


2.56 


20.83 


2.65 


20.82 


2.74 


20.81 


2.83 


22 


21.84 


2.68. 


21.82 


2.78 


21.81 


2.87 


21.80 


2.97 


22 


23 


22.83 


2.80'! 22.82 


2.90 


22.80 


3.00 


22.79 


3.10 


23 


24 


23.82 


2.92 


23.81 


3.03 


23.79 


3.13 


23.78 


3.24 


24 


25 


24.81 


3.05 


24.80 


3.15 


24.79 


3.26 


24.77 


3.37 


25 


26 


25.81 


8.17 


25.79 


3.28 


25.78 


3.39 


25.76 


3.51 


26 


27 


26.80 


3.29 


26.78 


3.41 


26.77 


3.52 


26.75 


3.64 


27 


28 


27.79 


3.41 


27.78 


3.53 


27.76 


3.65 


27.74 


3.78 


28 


29 


28.78 


8.53 


28.77 


3.66 


28.75 


3.79 


28.74 


3.91 


29 


30 


29.78 


3.06 


29.76 


3.79 


29.74 


3 92 


29.73 


4.05 


30 


31 


30.77 


3.78 


30.75 


3.91 


30.73 


4.05 


30.72 


4.18 


31 


32 


31.76 


3.90 


31.74 


4.04 


31.73 


4.18 


31.71 


4.32 


32 


33 


32.75 


4.02 


32.74 


4.16 


32.72 


4.31 


32.70 


4.45 ! 33 


34 


33.75 


4.14 


33.73 


4.29 


33.71 


4.44 


33.69 


4.58 ' 34 


35 


34.74 


4.27 


34.72 


4.42 


34.70 


4.57 


34.68 


4.72! 35 


36 


35.73 


4.39 


35.71 


4.54 


35.69 


4.70 


35.67 


4.85 i isC 


37 


36.72 


4.51 


36.70 


4.67 


36.68 


4.83 


36.66 


4.99 


37 


38 


37.72 


4.63 


37.70 


4.80 


37.67 


4.96 


37.65 


5.12 


38 


39 


38.71 


4.75 


38.69 


4.92 


38.67 


5.09 


38.64 


5.26 i 39 1 


40 
41 


39.70 


4.87 


39.68 


5.05 


39.66 


5.22 


39.63 


5.39 


40 


40.70 


5.00 


40.67 


5.17 


40.65 


5.35 


40.63 


5.53 


41 


42 


41.69 


5.12 


41.66 


5.30 


41.64 


5.48 


41.62 


5.66 


42 


43 


42.68 


5.24 


42.66 


5.43 


42.63 


5.61 


42.61 


5.80 


43 


44 43.67 


5.36 


43.65 


6.55 


43.62 


5.74 


43.60 


6.93 


44 


45 44.67 


5.4S 


44.64 


5.68 


44.62 


5.87 


44.59 


6.07 


45 


46 45.66 


5.61 


45.63 


5.81 


45.61 


6.00 


45 58 


6.20 


46 


47 46.65 


5.73 


46.62 


5.93 


46.60 


6.13 


46.57 


6.34 


47 


4« 


47.64 


5.85 


47.62 


6.06 


47.59 


6.27 


1 47.56 


6.47 


48 


49 


48.63 


5.97 


48.61 


6.18 


48.58 


6.40 


48.55 


6.61 


49 


60 


49.63 


6.09 


49.60 


6.31 


49.57 


6.63 


49.54 


6.74 


50 

e 



s 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


§ 


















1 














Q 


S3] 


Deg. 


82J 


Deg. 


82| 


Deg. 


82i 


Deg. 


.3 
1 



■f&AVEKSB TABLE. 



87 





7Deg. 


7iDeg. 


H Deg. 


7} Deg. 


a 


!1 

s 

9 

ir 




Dep. 




Dep. 
6.44 




\ 






s 

61 


L&t. 


Lat. 


Lat. 


Dep. 


Lat. 


Dep. 


60.62 


6.22 


50.59 


6( 56 


6.66 


.50.63 


6>8S 


62 


61.61 


6.34 


51.58 


6.56 


51.56 


6.79 


51.53 


7.01 


52 


63 


62.60 


6.46 


52.58 


6.69 


52.55 


6.92 


52.52 


7.16 


63 


64 


53.60 


6.58 


53.57 


6.81 


53.54 


7.05 


53.51 


7.28 


64 


65 


54.59 


6 70 


54.56 


6.94 


54.53 


7.18 


54.50 


r.42 


55 


56 


65.58 


6.82 


£5.55 


7.07 


55.52 


7 31 


55.49 


7.55 


56 


67 


66.58 


6.95 


56.54 


7.19 


56.51 


7 44 


56.48 


7.69 


57 


58 


57.57 


7.07 


57.54 


7.32 


57.50 


7.57 


67.47 


7.82 


58 


59 


58.56 


7.19 


58.53 


7.45 


58.50 


7.70 


58.46 


7.96 


59 


60 


59.55 


7.31 
7 43 


59.52 


7.57 


59.49 


7.83 


59.45 


8.09 


60 
"fil 


60.55 


60.51 


7.70 


60.48 


7.96 


60.44 


8.23 


62 


61.54 


7.56 


61.50 


7.82 


61.47 


8.09 


61.43 


8.36 


62 


63 


62.53 


7.68 


63.50 


7.95 


62.46 


8.22 


62.42 


8.50 


63 


64 


63.52 


7.80 


63.49 


8.08 


63.45 


8.35 


63.42 


8.63 


64 


65 


64.52 


7.92 


64.48 


8.20 


64.44 


8.48 


64.41 


8.77 


65 


66 


65.51 


8.04 


65.47 


8.33 


65.44 


8.61 


65.40 


8.90 


66 


67 


66 50 


8.17 


66.46 


8.46 


66.43 


8.75 


66.39 


9.04 


67 


68 


67.49 


8.29 


67.46 


8.58 


67.42 


8.88 


67.38 


9.17 


68 


69 


68.49 


8.41 


68.45 


8.71 


68.41 


9.01 


68.37 


9.30 


69 


70 
71 


69.48 
70.47 


8.63 


69.44 


8.83 


69.40 
70.39* 


9.14 


69.36 
70.35 


9.44 


70 
71 


8.65 


70.43 


8.96' 


9.27 


9.57 


72 


71.46 


8.77 


71.42 


9.09 


71.38 


9.40 


71.34 


9.71 


72 


73 


72.46 


8.90 


72.42 


9.21 


72.38 


9.53 


72.33 


9.84 


73 


74 


73.45 


9.82 


73.41 


9.34 


73.37 


9.66 


73.32 


9.9S 


74 


75 


74.44 


9.14 


74.40 


9.46 


74.36 


9.79 


74.31 


10.11 


75 


76 


75.43 


9.26 


75.39 


9.59 


75.35 


9.92 


75.31 


10.25 


76 


77 


76.43 


9.38 


76.38 


9.72 


76.34 


10.05 


76.30 


10.38 


77 


78 


77.42 


9.51 


77.38 


9.84 


77.33 


10.18 


77.29 


10.52 


78 


79 


78.41 


9.63 


78.37 


9.97 


78.32 


10.31 


78.28 


10.65 


79 


80 
81 


79.40 


9.75 


79.36 


10.10 


79.32 


10.44 


79.27 
80.26 


10.79 


80 
81 


80.40 


9.87 


80.35 


10.22 


80.31 


10.67 


10.92 


82 


81.39 


9.99 


81.34 


10.35 


81.30 


10.70 


81.25 


11.06 


82 


83 


82.38 


10.12 


82.34 


10.47 


82.29 


10.83 


82.24 


11.19 


83 


84 


83.37 


10.24 


83.33 


10.60 


83.28 


10.96 


83.23 


11.33 


84 


85 


84.37 


10.36 


84.32 


10.73 


84.27 


11.09 


84.22 


11.46 


85 


86 


85.36 


10.48 


85.31 


10.85 


85.26 


11.23 


85.21 


11.60 


86 


S7 


86.35 


10.60 


86.30 


10.98 


86.26 


11.36 


86.21 


11.73 


87 


38 


87.34 


10.72 


87.30 


11.11 


87.25 


11.49 


87.20 


11.87 


88 


89 


88.34 


10.85 


88.29 


11.23 


88.24 


11.62 


88.19 


12.00 


89 


90 
91 


89.33 


10.97 


89.28 


11.36 


89.23 


11.75 


89.18 


12.14 


90 
91 


90.32 


11.09 


90.27 


11.48 


90.22 


11.88 


90.17 


12.27 


92 


91.31 


11.21 


91.26 


11.61 


91.21 


12.01 


91.16 


12.41 


92 


93 


92.31 


11.33 


92.26 


11.74 


92.20 


12.14 


92.15 


12.54 


93 


94 


93 30 


11.46 


93.25 


11.86 


93.20 


12.27 


93.14 


12 68 


94 


95 


94.29 


11.58 


94.24 


11.99 


94.19 


12.40 


94.13 


12.81 


as 


96 


95.28 J 


11.70 


95.23 


12.12 


95.18 


12.53 


95.12 


12.95 


96 


S7 


96.28 


11.82 


96.22 


12.24 


96.17 


12.66 


96.11 


13.08 


97 


98 


97.27 


11.94 


97.22 


12.37 


97.16 


12.79 


97.10 


13.22 


98 


99 


98.26 


12.07 


98.21 


12.49 


98.15 


12.92 


98.10 


13.35 


99 


lao 

8 


99.25 


12.19 


99.20 


12.62 


99.14 


13.05 


99.09 


13.49 


:oo 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


i 


















i3 


















.9 


83 1 


)eg. 


821 


Deg. 


82| 


Deg. 


m 


Deg. 


5 



88 



TEAVBllSE TABLE. 



5 


8 Deg. 


8i Deg. 


6^ Deg. 


8J Deg. 


5 




















? 


















1 

1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 3 « 


0.99 


0.14 


0.99 


0.14 


0.99 


0.15 


0.99 


O.lo' 1 


2 


1.98 


0.28 


1.98 


0.29 


1.98 


0.30 


1.98 


0..30 a 


3 2.97 


0.42 


2.97 


0.43 


2.97 


0.44 


2.97 


0.46 3 


4 


3.96 


0.56 


3.96 


0.57i 


3.96 


0.59 


3.95 


0.61 i 


5 


4.95 


0.70 


4.95 


0.72 


4.95 


0.74 


4.94 


0.76, 5 


6 


5.94 


0.84 


5.94 


0.86 


5.93 


0.89 


5.93 


0.91 6 


7 


6.93 


0.97 


6.93 


1.00 


6.92 


1.03 


6.92 


1.06 7 


81 


7.92 


1. 11 


7.92 


1.15 


7.91 


1.18 


7.91 


1 22 S 


S 


8.91 


1.25 


8.91 


1.29! 


8.90 


1.33 


8.90 


1.37 


9 


10 

11 


9.90 
10.89 


1.39 1 

1.53 i 


9.90 


1.43! 


9.89 
10.88 


1.481 
1.63i 


9. 83 


1.52 
1.67 


10 
11 


10.89 


1.58 


10.87 


12 


11.88 


1.67; 


11.88 


1.72 


11.87 


1.77 


11.86 


1.83 


12 


13 


12.87 


1.81 ! 


12.87 


1.87 


12.86 


1.92 


12.85 


1.98 


13 


14 


13.86 


1.95 


13.86 


2.01 


13.85 


2.07 


13.84 


2.13 


14 


15 


14.85 


2.09 1 


14.85 


2.15 


14.84 


2.22! 


14.83 


2.28 


15 


16 


15.84 


2.23 


15.84 


2.30 


15.82 


2.36 1 


15.81 


2.43 


16 


17 


16.83 


2.37 


16.83 


2.44i 


16.81 


2.51 i 


16.80 


2.59 


17 


IS 


17.82 


2.51 


17.81 


2.58 1 


17.80 


2.66 


17.79 


2.74 


18 


19 


18.82 


2.64 


18.80 


2.73 


18.79 


2.81 


18.78 


2.89 


19 


20 

21 


19.81 


2.78 


19.79 


2.87 


19.78 


2.96 


19.77 


3.04 


20 


20.80 


2.92 


20.78 


3.01 


20.77 


3.10 20.76 


3.19 21 


22 


21.79 


3.06 


21.77 


3.16 


21.76 


3.25 21.74 


3.35 i 22 


23 


22.78 


3.20 


22.76 


3.30 


22.75 


3.40 22.73 


3.50 1 23 


24 


23.77 


3.34 


23.75 


3.44 


23.74 


3.55 23.72 


3.65 1 24 


25 


24.76 


3.48 


24.74 


3.59 


24.73 


3.70 24.71 


3.80 1 25 


26 


25.75 


3.62 


25.73 


3.73 


25.71 


3.84 25.70 


3.96 1 26 


27 


26.74 


3.76 


26.72 


3.87 26.70 


3.99 26.69 


4.11 27 


28 


27.73 


3.90 


27.71 


4.02 


1 27.69 


4.14| 27.67 


4.26 i 28 


29 


23.72 


4.04 


28.70 


4.16 


28.63 


4.29 1 28.66 


4.41 29 


30 


29.71 
30.70 


4.18 
4.31 


29.69 
30.68 


4.30 
4.45 


29.67 


4.43 '29.65 


4.56 i 30 


30.66 


4.53 30.64 


4.72 31 


32 


31.69 


4.45 


31.67 


4.59 


; 31.65 


4.73 : 31.63 


4.87 32 


33 132.68 


4.59 


32.66 


4.74 


i 32.64 


4.88 j 32.62 


5.02 33 


34 


33.67 


4.73 


33.65 


4.88 


33.63 


5.03 1 33.60 


5.17 1 34 


3o 


34.66 


4.87 


34.64 


5.02 


34.62 


5.17 34.59 


5.32 35 


36 


35.65 


5.01 


35.63 


5.17 


35.60 


5.32 35.58 


5.^8 36 


37 


36.64 


5.15 


36.62 


5.31 


36.59 


5.47 36.57 


5.63 1 37 


38 


37.63 


5.29 


37.61 


5.45 


37.58 


5.62 37.56 


5.78 38 


39 


38.62 


5.43 


38.60 


5.60 


38.57 


5.76 38.55 


5.93 39 


40 
41 


39.61 
40.60 


5.57 
5.71 


i 39-59 


5.74 


39.56 


5.91 
6.06 


39.53 


6.08 40 


i 40.58 


5.88 


40.55 


40.52 


6.24 41 


42 


41.59 


5.85 


,41.57 


6.03 


141.54 


6.21 |I41.51 


6.39 42 


43 


42.58 


5.99 


42.56 


6.17 


142.53 


6.36 42.50 


6.54 43 


44 


43.57 


6.12 


43.54 


6.31 


43.52 


6.50 43.49 


6.69! 44 


if, 


44- 56 


6.26 


44.53 


6.46 


44.51 


6.65 1 44.48 


6.85 


45 


46 


45.55 


6.40 


45.52 


6.60 


45.49 


6.80 t 45.46 


7.00 


46 


17,46.54 


6.54 


46.5 


6.74 


46.48 


6.95 1 46.45 


7.15 


47 


iS 


47.53 


6.68 


47.50 


6.89 


47.47 


7.09 47.44 


7.30 1 48 \ 


49 


48.52 


6.82 


48.49 


7.03 


48.46 


7.24 48.43 


7.45 


49 


60 

s 
s 


49.51 


6.96 


49.48 


7.17 


49.45 


7.39 49.42 


7.61 


50 


Dep. 
83 1 


Lat. 
deg. 


Dep. 
811 


Lat. 
Deg. 


Dep. 


Lat. 1 Dep. 


Lat, 


9 
U 

S 


8U 


Deg. 


m 


Deg. 


S 

.2 



TRAVERSE TABLE 



89 



c 

S' 

a 
o 

51 


3 Deg. 


8i 
LaL 


Deg. 


8iDeg. 


81 Deg. 


1 

1 

a 
9 

51 


Lat. 
50.50 


Dep. 

7.10 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


50.47 


7.32 


50.44 


7.54 


50.41 


7.76 


52 


51.49 


7.24 


51.46 


7.46 


51.43 


7.69 


51. .39 


7.91 


52 


63 


62.48 


7.38 


52.45 


7.61 


52.42 


7.83 


52.38 


8.06 


53 


54 


53,47 


7.52 


53.44 


7.75 


53 41 


7.98 


53.37 


8.21 


54 


55 


54.46 


7.65 


54.43 


7.89 


54,40 


8.13 


54.36 


8.37 


55 


56 


55.46 


7.79 


55.42 


8.04 


55.38 


8.28 


55.35 


8.52 


56 


57 


56.45 


7.93 


56.41 


8.18 


56.37 


8.43 


56.34 


8.67 


57 


58 


57.44 


8.07 


57.40 


8.32 


57.36 


8.57 


57.32 


8.82 


58 


59 


58.43 


8.21 


58.39 


8.47 


58.35 


8.72 


58.31 


8.98 


59 


(50 
61 


59.42 
60.41 


8.35 


59.38 


8.61 


59.34 


8.87 


59.30 


9.13 

"9.28 


60 
61 


8.49 


60.37 


8.75 


60.33 


9.02 


60.29 


62 


61.40 


8.63 


61.36 


8.90 


61.32 


9.16 


61.28 


9.43 


62 


63 


62.39 


8.77 


62.35 


9.04 


62.31 


9.31 


62.27 


9.58 


63 


64 


63.38 


8.91 


63.34 


9.18 


63.30 


9.46 


63.26 


9.74 


64 


65 


64.. 37 


9.05 


64.33 


9.33 


64.29 


9.61 


64.24 


9.89 


65 


66 


65.36 


9.19 


65.32 


9.47 


65.28 


9.76 


65.23 


10.04 


66 


67 


66.35 


9.32 


66.31 


9.61 


66.26 


9.90 


66.22 


10.19 


67 


68 


67.34 


9.46 


67.30 


9.76 


67.25 


10.05 


67.21 


10.34 


68 


69 


68.33 


9.60 


68.29 


9.90 


68.24 


10.20 


68.20 


10.50 


69 


70 
71 


69.32 


9.74 


69.28 


10.04 


69.23 


10.35 


69.19 


10.65 


70 

71 


70.31 


9.88 


70.27 


10.19 


70.22 


10.49 


70.17 


10.80 


72 


71.30 


10.02 


71.25 


10.33 


71.21 


10.64 


71.16 


10.95 


72 


73 


72.29 


10.16 


72.24 


10.47 


72.20 


10.79 


72.15 


11.10 


73 


74 


73.28 


10.30 


73.23 


10.62 


73.19 


10.94 


73.14 


11.26 


74 


75 


74.27 


10.44 


74.22 


10.76 


74.18 


11.09 


74.13 


11.41 


75 


78 


75.26 


10.58 


75.21 


10.91 


75.17 


11.23 


75.12 


11.56 


76 


77 


76.25 


10.72 


76.20 


11.05 


76.15 


11.38 


76.10 


11.71 


77 


78 


77.24 


10.86 


77.19 


11.19 


77.14 


11.53 


77.09 


11.87 


78 


79 


78.23 


10.99 


78.18 


11.34 


78.13 


11.68 


78.08 


12.02 


79 


80 
81 


79.22 


11.13 


79.17 


11.48 


79.12 


11.82 


79.07 


12.17 


80 
"81 


80.21 


11.27 


80.16 


11.62 


80.11 


11.97 


80.06 


12.32 


82 


81.20 


11.41 


81.15 


11.77 


81.10 


12.12 


81.05 


12.47 


82 


83 


82.19 


11.55 


82.14 


11.91 


82.09 


12.27 


82.03 


12.63 


83 


84 


83.18 


11.69 


83.13 


12.05 


83.08 


12.42 


83.02 


12.78 


84 


85 


84.17 


11.83 


84.12 


12.20 


84. 0-^ 


12.56 


84.01 


12.93 


85 


86 


85.16 


11.97 


85.11 


12.34 


85. 0(? 


12.71 


85.00 


13.08 


86 


87 


86.15 


12.11 


86.10 


12.48 


85.04 


12.86 


85.99 


13.23 


87 


88 


87.14 


12.25 


87.09 


12.63 


87.03 


13.01 


86.98 


13.39 


88 


89 


88.13 


12.39 


88.08 


12.77 


88.02 


13.16 


87.96 


13.54 


89 


90 

01 


89.12 


12.53 


89.07 


12.91 


89.01 


13.30 


88.95 
89.94 


13.69 


90 
91 


90.11 


12.66 


90.06 


13.06 


90.00 


13.45 


13.84 


92 


91.10 


12.80 


91.05 


13.20 


90.99 


13.60 


90.93 


14.00 


92 


' 93 


92.09 


12.94 


92.04 


13.34 


91.98 


13.75 


91.92 


14.15 


93 


94 


93.09 


13.08 


93.03 


, 13.49 


92.97 


13.89 


92.91 


14.30 


\H 


95 


94.08 


13.22 


94.02 


13.63 


93.96 


14.04 


93 89 


14.45 


i^5 


96 


95.07 


13.36 


95.01 


i 13.78 


94.95 


14.19 


94.88 


14.60 


96 


97 


96.06 


13.50 


96.00 


! 13.92 


95.93 


14.34 


95.87 


14.76 


97 


98 


97.05 


13.64 


96.99 


: 14.06 


96.92 


14.49 


96.86 


14.91 


98 


99 


98.04 


13.78 


97.98 


14.21 


97.91 


14.63 


97.85 


15.06 


99 


100 

6 

O 

a 
a 
*^ 
m 

a 


99.03 


13.92 


98.97 


14.35 


98.90 


14.78 


98.84 


15 21 


100 

s 

eO 

5 


Dep. 


Lat. 


Dtp. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


82 1 


Dog. 


811 De^. 


SHDeg. 


81i Dog. 



90 



TRAVtBSE TABLE 



s 

3 

a 


9 Deg. 


H Deg. 


H 


Deg 


9f 


Deg 


s 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 
0,99 


Dep. 


o 

2 


1 0.99 


0.16 


0.99 


0.16 


0.99 


0.17 


0.17 


I 


2 1.98 


0.31 


1.97 


0.32 


1.97 


0.33 


1.97 


0.34 


2 


3 


2.96 


0.47 


2.96 


0.48 


2.96 


0.60 


2.96 


0.51 


3 


4 


3.95 


0.63 


3.95 


0.64 


3.95 


0.66 


3.94 


0.68 


4 


5 


4.94 


0.78 


4.93 


0.80 


4.93 


0.83 


4.93 


0.86 


5 


6 


5.93 


0.94 


5.92 


0.96 


5.92 


0.99 


5.91 


1.02 


6 


7 


6.91' 1.10 


0.91 


1.13 


6.90 


1.16 


6.90 


1.19 


7 


8 


7.90 1.25 


7.90 


1.29 


7.89 


1.32 


7.88 


1.35 


8 


9 


8.89 1.41 


8.88 


1.45 


8.88 


'.49 


8.87 


'52 


9 


10 
11 


9.88 


1.56 


9.87 


1.61 


9.86 


X.65 


9.88 


1.69 


10 
11 


10.86 


1.72 


10.86 


1.77 


10.86 


1.82 


10.84 


1.86 


12 


11.85 


1.88 


11.84 


1.93 


11.84 


1.98 


11.83 


2.03 


12 


13 


12.84 


2.03 


12.83 


2.09 


12.82 


2.16 


12.81 


2.20 


13 


U 


13.83 


2.19 


13.82 


2.25 


13.81 


2.31 


13.80 


2.37 


14 


1.5 


14.82 


2.35 


14.80 


2.41 


14.79 


2.48 


14.78 


2.54 


16 


16 


16.80 


2.5C 


15.79 


2.57 


15.78 


2.64 


15.77 


2.71 


16 


17 


16.79 


2.66 


16.78 


2.73 


16.77 


2.81 


16.75 


2.88 


17 


18 


17.78 


2.82 


17.77 


2.89 


17.75 


2.97 


17.74 


3.05 


18 


19 


18.77 


2.97 


18.75 


3.05 


18.74 


3.14 


18.73 


3.22 


19 


20 

21 


19.75 


3.13 


19.74 


3.21 


19.73 


3.30 


19.71 


3.39 


20 
21 


20.74 


3.29 


20.73 


3.38 


20.71 


3.47 


20.70 


3.56 


22 


21.73 


3.44 


21.71 


3.64 


21.70 


3.63 


21.68 


3.73 


22 


28 


22.72 


3.60 


22.70 


3.70 


22.68 


3.80 


22.67 


3.90 


23 


24 


23.70 


3.75 


23.69 


3.86 


23.67 


3.96 


23.65 


4.06 


24 


25 


24.69 


3.91 


24.67 


4.02 


24.66 


4.13 


24.64 


4.23 


25 


2fi 


25.68 


4.07 


25.66 


4.18 


25.64 


4.29 


25.62 


4.40 


26 


27 


26.67 


4.22 


26.65 


4.34 


26.63 


4.46 


26.61 


4.. 57 


27 


28 


27.66 


4.38 


27.64 


4.50 


27.62 


4.62 


27.60 


4.74 


28 


29 


28.64 


4.54 


28.62 


4.66 


28.60 


4.79 


28.58 


4.91 


29 


30 


29.63 


4.69 


29.61 


4.82 


29.59 


4.95 


29.57 


6.08 


30 


31 


30.62 


4.85 


30.80 


4.98 


30.57 


5.12 


30.55 


6.25 


31 


32 


31.61 


5.01 


31.58 


5.14 


31.66 


5.28 


31.64 


6.42 


32 


33 


32.59 


5.16 


32.57 


5.30 


32.55 


5.46 


32.52 


6.59 


33 


34 


33.58 


5.32 


33.56 


5.47 


33.53 


5.61 


33.51 


5.76 


34 


35 


34.57 


5.48 


34.54 


5.63 


34.52 


5.78 


34.49 


6.93 


35 


36 


35.56 


5.63 


35.53 


5.79 


35.61 


5.94 


35.48 


6.10 


36 


37 


36.54 


5.79 


36.62 


6.96 


36.49 


6.11 


36.47 


6.27 


37 


38 


37.53 


5.94 


37.51 


6.11 


37.48 


6.27 


37.46 


6.44 


38 


39 


38.52 


6.10 


38.49 


6.27 


38.47 


6.44 


38.44 


6.60 


39 


40 
41 


39.51 


6.26 


39.48 


6.43 


39.45 


6.60 


39.42 


6.77 


40 
41 


40.50 


6.41 


40.47 


6.69 


40.44 


6.77 


40.41 


6.94 


42 


41,48 


6.57 


41.45 


6.75 


41.42 


6.92 


41.39 


7 11 


42 


43 


42 47 


6.73 


42.44 


6.91 


42.41 


7.10 


42.38 


7 28 


43 


44 


43,46 


6.88 


43.43 


7.07 


43.40 


7.26 


43.36 


7.45 


44 


V5 


44.45 


7.04 


1 44.41 


7.23 


44.38 


7.43 


44.35 


7.62 


45 


46 


45.43 


7.20 


' 45.40 


7.39 


45.37 


7.59 


45.34 


7.79 


16 


47 46.42 


7.35 


46.39 


7.55 


46.36 


7.76 


46.32 


7.96 


i7 


4t; 47 41 


7.51 


47.38 


7.72 


47.34 


7.92 


47.31 


8.13 


18 


49 48.40 


7.67 


48.36 


7.88 


48.33 


8.09 


48.29 


8.30 


49 


50' 


49.38 


7.82 


49.35 


8.04 


49.32 
Dep. 


8.25 


49.28 


8.47 


50 

s 


s 


Dep. 


Lat. 


Dep. 


Lat 


Lat. 


Dep. 


Lat. 


§ 















11 












.a 


81 Deg. 


801 


Dog. 


801 


Dog. 


8(H 


Deg. 


& 



TRAVERSE TABLE. 



91 



o 

6l 
52 
53 
64 
66 
56 
67 
58 
59 
60 

61 
62 
63 
64 
65 
66 
67 
68 
69 
70 

71 
72 
73 
74 
75 
76 
77 
78 
79 
80 

81 
82 
83 
84 
86 
86 
87 
88 
89 
90 

91 
92 
93 
94 
95 
96 
97 
98 
99 

too 

« 


9 Deg. 


1 

9^ Deg. 


H Deg. 


9| Deg 


g- 

p 

a 

? 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

6] 
62 
63 
64 
65 
66 
67 
68 
69 
70 

71 
72 
73 

74 
75 
76 
77 
78 
79 
80 

81 
82 
83 
84 
85 
86 
87 
88 
89 
90 

91 
32 
93 
94 
95 
96 
97 
98 
99 
iOO 

6 

ed 

Q 


Lat 

50,37 
61.36 
52.36 
53.34 
54.32 
55.31 
56.30 
57.29 
58.27 
59.26 

60.25 
61.24 
62.22 
63.21 
64.20 
65.19 
66.18 
67.16 
68.15 
69.14 


Dep. 


Lat. 

50.34 
51.32 
52.31 
53.30 
54<28 
65.27 
56.26 
67.25 
58.23 
59.22 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


7.98 
8.13 
8.29 
8.45 
8.60 
8.76 
8.92 
9.07 
9.23 
9.39 


8.20 
8.36 

8.62 
8.68 
8.84 
9.00 
9 16 
9.32 
9 48 
9 ^4 

9.81 
9.97 
10.13 
10.29 
10.45 
10.61 
10.77 
10.93 
11.09 
11.25 


50.30 
51.29 
52.27 
53.26 
54.25 
55.23 
56.22 
57.20 
68.19 
69.18 


8.42 
8.58 
8.75 
8.91 
9.08 
9.24 
9.41 
9.57 
9.74 
9.90 


60.26 
51.25 
52.23 
63.22 
54.21 
56.19 
56.18 
67.16 
68.15 
59.13 


8.64 
8.81 
8.98 
9.14 
9.31 
9.48 
9.66 
9.82 
9.99 
10.16 


9.. 54 
9.70 
9.86 
10.01 
10.17 
10.32 
10.48 
10.64 
10.79 
10.95 


60.21 
61.19 
62.18 
63.17 
64.15 
65.14 
66.13 
67.12 
68.10 
69.09 


■60.16 
61.15 
62.14 
63.12 
64.11 
65.09 
66.08 
67.07 
68.05 
69.04 


10.07 
10.23 
10.40 
10.56 
10.73 
10.89 
11.06 
11.22 
11.39 
11.55 


60.12 
61.10 
62.09 
63.08 
64.06 
5.05 
66.03 
67.02 
68.00 
68.99 


10.33 
10.60 
10.67 
10.84 
11.01 
11.18 
11.35 
11.52 
11.69 
11.85 


70.13 
71.11 
72.10 
73.09 
74.08 
75.06 
76.05 
77.04 
78.03 
79.02 
80.00 
80.99 
81.98 
82.97 
83.95 
84.94 
86.93 
86.92 
87.90 
88.89 


11. li 
11.26 
11.42 
11.58 
11.73 
11.89 
12.05 
12.20 
12.36 
12.61 


70.08 
71.06 
72.05 
73.04 
74.02 
75.01 
76.00 
76.99 
77.97 
78.96 


11 41 
11.57 
11.73 
11.89 
12.06 
12.22 
12.38 
12.. 54 
12.70 
12.86 


70.03 
71.01 
72.00 
72.99 
73.97 
74.96 
75.94 
76.93 
77.92 
78.90 


11.72 
11.88 
12.05 
12.21 
12.38 
12.54 
12.71 
12.87 
13.04 
13.20 


69.97 
70.96 
71.95 
72.93 
73.92 
74.90 
76.89 
76.87 
77.86 
78.84 


12.02 
12.19 
12.36 
12.53 
12.70 
12.87 
13.04 
13.21 
13.38 
13.55 


12.67 
12.83 
12.98 
13.14 
13.30 
13.45 
13.61 
13.77 
13.92 
14.08 
14.24 
It. 39 
14.65 
14.70 
14.86 
15.02 
16.17 
16.33 
16.49 
16.64 


79.95 
80.93 
81.92 
82.91 
83.89 
84.88 
85.87 
86.86 
87.84 
88.83 

89.82 
90.80 
91.79 
92.78 
d3.76 
94.75 
95.74 
96.73 
97.71 
98.70 


13.02 
13.18 
13.34 
13.50 
13.66 
13.82 
13.98 
14.16 
14.31 
14.47 


79.89 
80.88 
81.86 
82.85 
83.83 
84.82 
86.81 
86.79 
87.78 
88.77 


13.37 
13.53 
13.70 
13.86 
14.03 
14.19 
14.36 
14.62 
14.69 
14.86 


79.83 
80.82 
81.80 
82.79 
83.77 
84.76 
86.74 
86.73 
87.71 
88.70 


13.72 
13.89 
14.06 
14.23 
14.39 
14.56 
14.73 
14.90 
15.07 
16.24 


89.88 
90.87 
91. 8t 
92.84 
93.83 
94.82 
95.81 
96.79 
97.78 
98.77 


14.63 
14.79 
14.95 
16.11 
15.27 
15.43 
16.59 
16.76 
16.91 
16.07 


89.76 
90.74 
91.72 
92.71 
93.70 
94.68 
95.67 
96.66 
97.64 
98.63 


15.02 
16.18 
16.36 
16.61 
15.68 
15.84 
16.01 
16.17 
16.34 
16.50 


89.69 
90 67 
91.66 
92.64 
93.63 
94.61 
95.60 
96.68 
97.67 
98.66 


16.41 
16.58 
15.75 
15.92 
16.09 
16.26 
16 43 
16.60 
16.77 
16.93 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 


lat. 


1 
81 Deg. i 

i 


801 Deg. 


80^ 


Deg. 


m Deg. 



92 



FRAVERSE TABLE 



P 


10 Deg. 


m Deg. 


1 
10| Deg. 


|0| Deg. 


C 
5' 

5 

P 
1 


Lat. 


Dep. 

T.JT 


Lat. 


Dep. 


Lat j 
0.98 


Dep. 1 
0.18 


Lat. 

"orgs' 


D€p. 


1 0.98 


0.98 


0.18 


0.19 


2 1.97 


0.35 1.97 


0.30 


1.97' 


0.36 


1.96 


0.37 


% 


3 , 2.95 


0.52 2.95 


0.53 


2.95 


0.55 


2.95 


56 


3 


4! 3.94 


0.69 3,94 


0.71 , 


3.93, 


0.73 


3.93 


0.75 


4 


5 4 . 92 


0.87 1 4.92 


0.89 


4.92' 


0.91 


4.91 


0,93 


5 


6 5.91 


1.04 1 5.90 


1.07 


5.90 


1.09 


5.89 


i.l2 


6 


7 6.89 


1.22 1 6.89 


1.25 


6.88, 


1 28 


S.88 


I 31 


7 


8 7. 88 


1.39; 7.87 


1.42 


7.87' 


1 46 


?.86 


1.49 


8 


9 8.86 


1.56! 8.86 


1.60 


8.85 


1.84. 


8.84 


1.68 





10 9.85 

11 10 83 


1.74 1 9.84 
1.91 10.82 


1.78 
1.96 


9.83; 
10.82 1 


1.82' 
2.00 


9.82 


1.87 
2.05 


lO 
11 


10.81 


12 11 82 


2.08 : 11.81 


2.14 


11.80' 


2.19 


11.79 


2.24 


12 


13 12.80 


2.26 12.79 


2.31 


12.78 


2.37 


12.77 


2.42 


13 


14 13.79 


2; 43 i 13.78 


2.49 i 


13.77 


2 . 55 


13.75 


2.61 


14 


15 


14.77 


2.60 i| 14.76 


2.67' 


14.75 


2 . 73 


14.74 


2.80 


Vj 


16 


15.76 


2.78 '\ 15.74 


2.85 1 


15.73 


2.92 i 


15.72 


2.98 


16 


17 


16.74 


2.95 : 16.73 


3.03 1 


16.72 


3.10 1 


16.70 


3.17 


17 


18 


17.73 


3.13 ; 17.71 


3.20 


17.70 


3.28' 


17. e8 


3.36 


18 


19 


18.71 


3.30 ; 18.70 


3.38 


18.68 


3.46 


18.67 


3.54 


19 


20 


19.70 


3.47 19.68 


3.56 


19.67 


3.64, 


19.65 


3.73 


20 


21 


20.68 


3.65 20.66 


3.74 


20.65 


3.83 


20.63 


3.92 


21 


22 21.67 


3. 82 21.65 


3.91 


21.63 


4.01 


21.61 


4.10 


22 


23 


22.65 


3.99 22.63 


4.09 


22.61 


4.19 


22.60 


4.29 


23 


24 


23.64 


4.17 " 23.62 


4.27 


23 . 60 


4.37 


23 . 58 


4.48 


24 


25 


24.62 


4.34 24.60 


4.45 


24.58 


4.56 


24.56 


4.66 


25 


26 


25.61 


4.51 25.59 


4.63 


25.56 ; 


4.74 


25.54 


4.85 


26 


27 


26.59 


4.69 26.57 


4.80 


26.55 


4.92 


26.53 


5.04 


27 


28 


27.57 


4.86 27.55 


4.98 


27.53 i 


5.10 


27.51 


5.22 


28 


29 


28.56 


5.04 2S.54 


5.16 


28.51 i 


5.28 


28.49 


5.41 


29 


30 
31 


29 . 54 
30.53 


5.21 
5.38 


29.52 


5.34 


29.50 ! 
80.48, 


5.47 
5.65 


; 29.47 


5.60 


30 


30.51 


5.52 


30.46 


5.78 


31 


32 


31.51 


5.56 31.49 


5.69, 


31.46 


5.83 


31.44 


5.97 


32 


33 


32.50 


5.73 32.47 


5.87! 


32.45 


6.01 


32.42 


6.16 


33 


34 


33.48 


5.90 33.46 


6.051 


33.43 


6.20 


33.40 


6.34 


34 


35 


J4 4" 


6.03 ; 34.44 


6 . 23 


34.41 


6.38 


34.39 


6.53 


35 


36 


35.45 


6.25 35.43 


6.41 


35.40 ; 


6.56 


35.37 


6.71 


38 


; 37 


36.44 


6.42 36.41 


6.58 


36.38: 


6.74 


36.35 


6.90 


37 


38 


37.42 


6.60 37.39 


6.76 


37.36 


6.92 


37.33 


7.09 


38 


39 


38.41 


6.77 38.38 


6.94 


38.35 


7,11 


38.32 


7.27 


39 


1 40 
41 


39.39 
40.38 


6.95 39.36 
7.12 40.35 


7.12 
7.30 


39.33 
40.31 


7.29 
7.47 


39.30 


7.46 


40 
41 


40.28 


7.65 


42 


41.36 


7.29 41.33 


7.47 


41.30 


7.65 


41.26 


7.83 


42 


43 42.35 


7.47 42.31 


7.65 


42.28 


7.84 


142.25 


S.02 


43 


44 43.33 


7.64 43.30 


7.83 


43.26 


8.02 


1 43.23 


8.21 


44 


45 ,44.32 


7.81 44.28 


8.01 


44. 2f 


8.20 


44.21 


8.39 


45 


46 45.30 


7.99 45.27 


8.19, 


45.23 


8.38 


45.19 


8.58 


46 


47 46.29 


8.16 46.25 


. 8.36 


46.21 


8.57 


i46.l8 


8.77 


47 


48 47.27 


8.34 


47.23 


8.54 


47.20 


8.75 


47.16 


8.95 


48 


43 48.26 


8.51 


48.22 


; 8.72 


48.18 


8.93 


48.14 


9.14 


49 


' 6<» 49.24 


8.68 
Lat. 


49.20 
1 Dep. 


8.90 


49.16 


9.11 
Lat. 


49.12 


1 9.33 
Lat. 


50 

o 

1 


i 

S 


l>ep. 


Lut. 


Dep. 


Dep. 


■mi 














'5 so 1 


[)eg. 


m 


Deg. 


791 


Deg. 


791 


Deg 


Q 


, 1 


^O^X^B^ 


1 















TKAVERSE TABLE. 



93 



Dista 


10 Deg, 


lOi Deg. 


lOi Deg. 


101 Deg. 


O 

5- 

3 
? 

5J 


3 
P 


Lat. 


Dep. 


Lat. 


Dep. 

9.08 


Lat. 


Dep. 


Lat. 


Dep, 


51 


50.23 


8.86 


50.19 


50.15 


9.29 


50 J 


9.51 


52 


51.21 


9.03 


51.17 


9.25 


51.13 


9.48 


61.09 


9 70 


52 


53 


52.19 


9.20 


52.15 


9.43 


.52.11 


9.66 


.52.07 


9.89 


53 


54 


53.18 


9.38 


53.14 


9.61 


53.10 


9.84 


63 . 05 


10.07 


64 


55 


54.16 


9.55 


54.12 9.79 


54.08 


10.02 


04.03 


10.26 


5.'i 


56 


55.15 


9.72 


55.111 i' 96 


55.06 


10.21 


55.02 


10.45 


56 


57 


56.13 


9.90 


56.09! 10.14 


56 . 05 


10.39 


56.00 


10.63 


57 


58 


57.12 


10.07 


57.07 1 10.32 


57 . 03 


10.57 


.56 . 98 


10.82 


58 


69 


58.10 


10.25 


58.06 ! 10.50 


58.01 


10.75 


67.96 


11.00 


69 


60 
61 


59.09 
60.07 


10.42 
10.59 


59.04! 10.68 


59.00 


10.93 


68.95 


11.19 


60 
'61 


60.03 10.85 


59.98 


11.121 


69.93 


11.38 


62 


61.06 


10.77 


61.01 ; 11.03 


60.96 


11.30 


60.91 


11.56 


62 


63 


62.04 


10.94 


61.99 i 11.21 


61.95 


11.48 


61.89 


11.75 


63 


84 


63.03 


11.11 


62.98 ! 11.39 


62.93 


11.66 


6S.88 


11.94 


64 


65 


64.01 


11.29 


63.96 i 11.57 


63.91 


11.85 


63.86 


12.12 


65 


66 


65.00 


11.46 


64.95 


11.74 


64.89 


12.03 


64.84 


12.31 


66 


67 


65.98 


11.63 


65.93 


11.92 


65.88 


12.21 


65.82 


12.60 


67 


68 


66.97 


11.81 


66.91 


12.10 


66.86 


12.39 


66.81 


12.68 


68 


69 


67.95 


11.98 


67.90 


12.28 


67.84 


12.57 


67.79 


12.87 


69 


70 
71 


68.94 
"69.92 


12.16 
12.33 


68.88 


12.46 


68.83 
69.81 


12.76 


68.77 
69.76 


13.06 


70 

7] 


69.87 


12.63 


12.94 


13.24 


72 


70.91 


12.50 


70.85 : 12.81 


70.79 


13.12 


70.74 


13.43 


72 


73 


71.89 


12.68 


71.83 12.99 


71.78 


13.30 


71.72 


13.62 


73 


74 


72.88 


12.85 


72.82 13.17 


72.76 


13.49 


'72.70 


13.80 


74 


75 


73.86 


13.02 


73.80 : 13.35 


73.74 


13.67 


73.68 


13.99 


76 


76 


74.85 


13.20 


74.79 13.52 


74.73 


13.85 


74.67 


14.18 


76 


77 


75.83 ! 13.37! 


75.77! 13.70 


75.71 


14.03 


76.65 


14.36 


77 


78 


76.82 1 13.54 


76.76 i 13.88 


76.69 


14.21 


76.63 


14.65 


78 


79 


77.80 1 13.72 


77.74 14.06 


77.68 


14.40 


77.61 


14.74 


79 


80 
81 


78.78 


13.89 
14.07 


78.72 


14.24 
14.41 


78.66 


14.58 


78.60 


14.92 


80 
81 


79.77 


79.71 


79.64 


14.76 


79.58 


15.11 


82 


80.75 


14.24 


80.69 , 14.59 


80.63 


14.94 


80.50 


15.29 


82 


83 


81.74 


14.41 


81.68 14.77 


81.61 


15.13 


81.54 


15.48 


83 


84 


82.72 


14.59 


82.66 14.95 


82.59 


15.31 


82.63 


15.67 


84 


85 


83.71 i 14.76 


83.64 15.13 


83.58 


15.49 


83.51 


16.85 


85 


86 


84.09 14.93 


84.63 15.30 


84.56 


15.67 


84.49 


16.04 


86 


87 


85.68 15.11 


85.61 , 15.48 


85.54 


15.85 


85.47 


16.23 


87 


88 


86.66 i 15.28 


86.60 15.66 


86.53 


16.04 


83.46 


16.41 


88 


89 


87.65' 15.45 


87.58 1 15.84 


87.51 


16.22 


87.44 


16.60 


89 


90 
91 


88.63 1 15.63 


88.56 


j 16.01 
16.19 


88.49 


16.40 


88.42 


16.79 
16.97 


90 

i'^91 


89.62] 15.80 


89.55 


89.48 


[16.58 


89.40 


92 


90.60 1 15.98 


90.53 16.37 


90.46 


16.77 


90.39 


17.16 


92 


93 


91.59 1 16.15 


91.52 16.55 


91.44 


16.95 


91.37 


17.35 


53 


94 


92.57 16.32 


92.50, 16.73 


92.43 


17.13 


92.35 


17.53 


94 


95 


93.56 16.50 


93.48 16.90 


93.41 


17.31 


93.33 


17.72 


95 


96 


'94.54 16.67 


94.47 17.08 


94.39 


17.49 


94.32 


17.91 


96 


97 95- 63 16.84 


95.45 17.26 


95.38 


17.68 


95.30 


18.09 


1 97 


98 96.51 17.02 


96.44, 17.44 


96.36 


17.86 


96.28 


18.28 


! 98 


99 


97.50 17.19 


97.42' 17.62 


97.34 


18.04 


97.26 


18.47 


! 99 


100 

o 

§ 
ca 


98.48 
Dep. 


I 17.36 


98.40 i 17.79 


98.33 


18.22 


98.25 


18.65 


! .00 

6 

go 

1. Q 


Lat. 


Dep. i Lat. 


Oep. 


Lat. 


Dep. 


Lat. 


1 80 Deg. 


791 Deg. 

i 


79i Deg. 


79i Deg. 



94 



TRAVERSE TABLE. 



1 

s 

• 


11 Deg. 


lU Deg. 


IH 


Deg. 


Ill Deg. 


■ 
B 

s 

1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 
0.20 


Lat. 


Dep. 


1 


0.98 


0.19 


0.98 


0.20 


0.98 


0.98 


0.20 


2 1.96 


0.38 


1.96 


0.39 


1.96 


0.40 


1.96 


0.41 


3 


3| 


2.94 


0.57 


2.94 


0.59 


2.94 


0.60 


2.94 


0.61 


a 


1 


3.93 


0.76 


3.92 


0.78 


3.92 


0.80 


3.92 


0.82 


4 


5 


4.91 


0.95 


4.90 


0.98 


490 


1.00 


4.90 


1.02 


5 


€ 5.89 


1.14 


6.88 


1.17 


6 88 


\ .20 


5.87 


1.22 


ft 


7 6.87 


1.34 


6.87 


1.37 


6.86 


1.40 


6.85 


1.43 


7 


8 


7.85 


1.53 


7.85 


1.56 


7.84 


1.59 


7.83 


1.63 


8 


9 


8.83 


1.72 


8.83 


1.76 


8.82 


1.79 


8.81 


1.83 


9 


10 
11 


9.82 


1.91 


9.81 
10.79 


1.95 


9.80 


1.99 


9.79 


2.04 


10 
11 


10.80 


2.10 


2.15 


10.78 


2.19 


10.77 


2.24 


12 


11.78 


2.29 


11.77 


2.34 


11.76 


2.39 


11.75 


2.44 


12 


13 


12.76 


2.48 


12.75 


2.54 


12.74 


2.59 


12.73 


2.66 


13 


14 


13.74 


2 67 


13.73 


2.73 


13.72 


2.79 


13.71 


2.86 


14 


15 


14.72 


2,86 


14.71 


2.93 


14.70 


2.99 


14.69 


3.06 


15 


16 


15.71 


3.05 


15.69 


3.12 


15.68 


3.19 


15.66 


3.26 


16 


17 


16.69 


3.21 


16.67 


3.32 


16.66 


3.39 


16.64 


3.46 


17 


18 


17.67 


3.43 


17.65 


3.51 


17.64 


3.59 


17.62 


3.66 


18 


19 


18.65 


3.63 


18.63 


3.71 


18.62 


3.79 


18.60 


3.87 


19 


20 
21 


19.63 


3.82 


19.62 


3.90 


19.60 


3.99 


19.58 


4.07 


20 
21 


20.61 


4.01 


20.60 


4.10 


20.58 


4.19 


20.56 


4.28 


22 


21.60 


4.20 


21.58 


4.29 


21.56 


4.39 


21.54 


4.48 


22 


23 


22.58 


4.39 


22.56 


4.49 


22.54 


4.. 59 


22.52 


4.68 


II 


24 


23.56 


4.58 


23.54 


4.68 


23.52 


4.78 


23.50 


4.89 


25 


24.54 


4.77 


24.62 


4.88 


24.50 


4.98 


24.48 


5.09 


25 


26 


25.52 


4.96 


25.50 


5.07 


25.48 


5.18 


25.46 


5.30 


26 


27 


26.50 


5.15 


26.48 


5.27 


26.46 


5.38 


26.43 


5.50 


27 


28 


27.49 


6.34 


27.46 


5.46 


27-44 


5.58 


27.41 


5.70 


28 


29 


28.47 


5.53 


28.44 


5.66 


28.42 


5.78 


28.39 


5.91 


29 


30 
31 


29.45 


5.72 


29.42 


5.85 


29.40 


5.98 


29.37 


6.11 


30 


30.43 


5.92 


30.40 


6.05 


30.38 


6.18 


30.35 


6.31 


31 


32 


31.41 


6.11 


31.39 


6.24 


31.36 


6.38 


31.33 


6.52 


32 


33 


32.39 


6 30 


32.37 


6.44 


32.34 


6.58 


32.31 


6.72 


33 


34 


33.38 


6.49 


33.35 


6.63 


33.32 


6.78 


33.29 


6.92 


34 


35 


34.36 


6.68 


34.33 


6.83 


34.30 


6.98 


34.27 


7.13 


35 


36 


35.34 


6.87 


35.31 


7.02 


35.28 


7.18 


35.25 


7 33 


36 


37 


36.32 


7.06 


36.29 


7.22 


36.26 


7.38 


36.22 


r.53 


37 


38 


37.30 


7.25 


37.27 


7.41 


37.24 


7.58 


37,20 


7.74 


38 


39 


38.28 


7.44 


38.25 


7.61 


38.22 


7.78 


38.18 


7.94 


39 


40 


39.27 


7.63 


39.23 


7.80 


39. 20^ 


7.97 


39.16 
40.14 


8.15 


40 
41 


41 


40.25 


7.82 


40.21 


8.00 


40.18 


8.17 


8.35 


42 41.23 8.01 


41.19 


8.19 


41.16 


8.37 


41.12 


8.56 


45 


43,42.21 8.20 


42.17 


8.39 


42.14 


8.57 


42.10 


8.76 


43 


41 ; 43. 19 8.40 


43.15 


8.58 


43.12 


8.77 


43.08 


8.9G 


4) 


45 1 44 17 1 8.59 


44.14 


8.78 


44.10 


8.97 


44.06 


9.16 


45 


46*45.15 8.78 


45.12 


8.97 


45.08 


9.17 


45.04 


9.37 


46 


47 146.14 8.97 


46.10 


9.17 


46.06 


9.37 


46.02 


9.57 


47 


48 j 47.12 9.16 


47.08 


9.36 


47.04 


9.57 


46.99 


9.78 


18 


49 148.10 9.35 


48.06 


9.50 


48.02 


9.77 


47.97 


9.98 


49 


50^ 


149.08 


9.54 


49.04 
Dep. 


9,75 


49.00 


9.97 


48.95 


10.18 


50 

' 8 

a 

ei 


Dep. 


Lat. 

1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


79] 


Deg 


78J De^. 


'H 


Deg. 


78i 


Deg. 



T1«A VERSE TAWLE. 



95 



o 


U Deg. 


lU Deer. 


IH Deg. 


Ill Deg. 


O 


a. 

Pi 

3 


















~~ 










n 
51 


LaU 


Dep. 


Lat. 


C/ep. 
' 9.95 


Lat. 


Dep. 


Lat. 


Dep. 


51 


50.06 


9.73 


60.02 


49.98 


10.17 


49.93 1 10.39 


52 


51.04 


9.92 


51.00 


10.14 


50.96 


10.37 


50.91 i 10.59 


62 


53 


52.03 


10.11 


51 98 


10.34 


51.94 


10.57 


51.89 


10.79 


53 


5'i 


53.01 


10.30 


52.96 


10.. 53 


52.92 


10.77 


62.87 


11.00 


54 


55 


53.99 


10.49 


53.94 


10.73 


53.90 


10.97 


63.85 


11.20 


65 


56 


64.97 


1 10.69 


54.92 


10.93 


64.88 


11.16 


54.83 


11.40 


56 


57 


55.95 


1 10,88 


55.90 


11.12 


55.86 


11.36 


55.81 


11.61 


5? 


58 


56.93 


1 11.07 


56.89 


11.32 


56.84 


11.56 


56.78 


11.81 


68 


59 


57.92 


11.26 


57.87 


11.51 


57.82 


11.76 


67.76 


12.01 


69 


60 

61 


58.90 
59.88 


11.45 


.58.85 


11.71 


58.80 


11.96 


58.74 


12.22 
12.42 


60 
61 


11.64 


59.83 


11.90 


59.78 


12.16 


69.72 


62 


60.86 


11.83 


60.81 


12.10 


60.76 


12.36 


60.70 


12.63 


62 


63 


61.84 


12.02 


61.79 


12.29 


61.74 


12.56 


61.68 


12.83 


63 


64 


62.82 


12.21 


62.77 


12.49 


62.72 


12.76 


62.66 


13.03 


64 


65 


63.81 


12.40 


63.75 


12.68 


63.70 


12.96 


63.64 


13.24 


66 


66 


64.79 


1 12.59 


64.73 


12.88 


64.68 


13.16 


64.62 


13.44 


66 


67 


65.77 


12.78 


65.71 


13.07 


65.66 


13. J6 


65.60 


13.64 


67 


68 


66.75 


12.98 


66.69 


13.27 


66.63 


13.56 


66.68 


13.85 


68 


69 


67.73 


13.17 


67.67 


13.46 


67.61 


13.76 


67.65 


14.05 


69 


70 
71 


68.71 


13.36 


68.66 


13.66 


68.69 


13.96 
14.16 


68.63 


14.25 


70 
"71 


69.70 


13.55 


69.64 


13.85 


69.57 


69.61 


14.46 


72 


70.68 


13.74 


70.62 


14.05 


70.55 


14.35 


70.49 


14.66 


72 


73 


71.66 


13.93 


71.60 


14.24 


71.53 


'4.55 


71.47 


14.87 


73 


74 


72.64 


14.12 


72.58 


14.44 


72.51 


14.75 


72.45 


16.07 


74 


75 


73.62 


14.31 


73.56 


14.63 


73.49 


14.95 


73.43 


15.27 


76 


76 


74.60 


14.50 


74.54 


14.83 


74.47 


15.15 


74.41 


16.48 


76 


77 


75.59 


14.69 


75.52 


15.02 


75.45 


15.35 


76 39 


15.68 


77 


78 


76.57 


14.88 


76.50 


15.22 


76.43 


15.55 


76.37 


15.88 


78 


79 


77.55 


15.07 


77.48 


15.41 


77.41 


15.76 


77.34 


16.09 


79 


80 
81 


78.53 


15.26 


78.46 


15.61 


78.39 
79.37 


16.95 


78.32 


16.29 


80 
"81" 


79.51 


15.46 


79.44 


15.80 


16.15 


79.30 


16.49 


82 


80.49 


15.65 


80.42 


16.00 


80.35 


16.35 


80.28 ' 16.70 


82 


83 


81.48 


15.84 


81.41 


16.19 


81.33 


16.55 


81.26 


16.90 


83 


84 


82.46 


16.03 


82.39 


16.39 


82.31 


16.75 


82.24 


17.11 


84 


85 


83.44 


16.22 


83.37 


16.58 


83.29 


16.95 


83.22 


17.31 


86 


86 


84.42 


16.41 


84.35 


16.78 


84.27 


17.15 


84.20 


17.51 


86 


87 


85.40 


16.60 


85.33 


16.97 


85.25 


17.35 


86.18 


17.72 


87 


88 


86.38 


16.79 


86.31 


17.17 


86.23 


17.64 


86.16 


17.92 


88 


89 


87.36 


16.98 


87.29 


17.36 


87.21 


17.74 


87.14 


18.12 891 


90 
91 


88.35 


17.17 


88.27 
89.25 


17.56 


88.19 


17.94 


88.11 


18.33 


901 


89.33 


17.36 


17.75 


89.17 


18.14 


89.09 


18.53 


91 


92 


90.31 


17.55 


90.23 


17.95 


90.15 


18.34 


90.07 


18.74 92 


93 


91.29 


17.75 


91.21 


18.14 


91.13 


18.64 


91.05 


18.94 93 


94 


92.27 


17.94 


92.19 


18.34 


92.11 


18.74 


92.03 


19.14 : 94 


95 


93.25 1 


18.13 


93.17 


18.53 


93.09 18.94 


93.01 


19.35 


95 


96 


94.24 


18.32 


94.16 


18.73 


94.07 .9.14 


93.99 


19.55 


96 


37 


95.22 


18.51 


95.14 


18.92 


95.05 19.34 


94.97 


19.75 


97 


98 


96.20 


18.70 


96.12 


19.12 


96.03 


19.64 


95.95 


19.96 


98 


99 


97.18 


18.89 


97.10 


19.31 


97.01 


19.74 


96.93 


20.16 


99 


100 

.3 
Q 


98.16 


19.08 


98.08 


19.61 


97.99 
Dep. 


19.94 


97.90 


20.36 


100 


Dop. 


Lat. 


Dep. 


Lat. 


Lat. 


Dep. 


Lat. 


.a ', 


79 D 


eg. 


II 
781 Deg. 


78i Deg. 


rSi Deg. 



96 



TRAVERSE TABLE 



5' 


12 Deg 


12i Deg. 


12^ 


Deg. 


12^ Deg. 


q 
8 


i 

• 


Lat. 


Dep. 

~oT2r 


Lat. 


Dep. 


Lat, 


Dep. 


Lat. 


DepL * 


1 


0.98 


0.98 


0.21 


0.98 


0.22 


0.98 


0.22 1 


2 


1.96 


0.42 


1.95 


0.42 


1.95 


0.43 


1.95 


0.44 « 


3 


2.93 


0.62 


2.93 


0.64 


2.93 


0.65 


2.93 


0.66 3 


4 


3.91 


0.83 


3.91 


0.85 


3.91 


0.87 


3.90 


0.88 4 


5 


4.89 


1.04 


4.89 


1.06 


4.88 


1.08 


4.88 


1.10 6 


6 


5.87 


1.25 


5.86 


1.27 


5.86 


1.30 


5.85 


1.32 3 


7 


6.85 


1.46 


6.84 


1.49 


6.83 


1.52 


6.83 


1 54 


7 


8 


7.83 


1.66 


7.82 


1.70 


7.81 


1.73 


7.80 


1 77 


8 


9 


8.80 


1.87 


8.80 


1.91 


8.79 


1.95 


8.78 


1.99 


9 


10 
11 


9.78 


2.08 


9.77 


2.12 


9.76 


2.16 
2.38 


9.75 
10.73 


2.21 
2.43 


10 

11 


10.76 


2.29 


10.75 


2.33 


10.74 


12 


11.74 


2.49 


11.73 


2.55 


11.72 


2.60 


11.70 


2.65 


12 


13 


12.72 


2.70 


12.70 


2.76 


12.69 


2.81 


12.68 


2.87 


13 


14 


13.69 


2.91 


13.68 


2.97 


13.67 


3.03 


13.65 


3.09 


14 


15 


14.67 


3.12 


14.66 


3.18 


14.64 


3.25 


14.63 


3.31 


15 


16 


15.65 


3.33 


15.64 


3.39 


15.62 


3.46 


15.61 


3.53 


16 


17 


16.63 


3.53 


16.61 


3.61 


16.60 


3.68 


16.58 


3.75 


17 


18 


17.61 


3.74 


17.59 


3.82 


17.. 57 


3.90 


17.56 


3.97 


18 


19 


18.58 


3.95 


18.57 


4.03 


18.55 


4.11 


18.53 


4.19 


19 


20 
21 


19.56 


4.16 


19.54 


4.24 


19.53 


4.33 


19.51 


4.41 


20 


20.54 


4.37 


20.52 


4.46 


20.50 


4.55 


20.48 


4.63 


21 


22 


21.52 


4.57 


21.50 


4.67 


21.48 


4.76 


21.46 


4.86 


22 


23 


22.50 


4.78 


22.48 


4.88 


22.45 


4.98 


22.43 


5.08 


23 


24 


23.48 


4.99 


23.45 


5.09 


23.43 


5.19 


23.41 


5.30 


24 


25 


24.45 


5.20 


24.43 


5.30 


24.41 


5.41 


24.38 


5.52 


25 


26 


25.43 


5.41 


25.41 


5.52 


25.38 


5.63 


25.36 


5.74 


26 


27 


26.41 


5.61 


26.39 


5.73 


26.36 


5.84 


26.33 


5.96 


27 


28 


27.39 


5.82 


27.36 


5.94 


27.34 


6.06 


27.31 


6.18 


28 


29 


28.37 


6.03 


28.34 


6.15 


28.31 


6.28 


28.28 


6.40 


29 


30 


29.34 


6.24 


29.32 


6.37 


29.29 


6.49 


29.26 


6.62 


30 
31 


31 


30.32 


6.45 


30.29 


6.58 


30.27 


6.71 


30.24 


6.84 


32 


31.30 


6.65 


31.27 


6.79 


31.24 


6.93 


31.21 


7.06 


32 


33 


32.28 


6.86 


32.25 


7.00 


32.22 


7.14 


32.19 


7.28 


33 


34 


33.26 


7.07 


33.23 


7.21 


33.19 


7.36 


33.16 


7.50 


34 


35 


34.24 


7.28 


34.20 


7.43 


34.17 


7.58 


34.14 


7.72 


35 


36 


35.21 


7.48 


35.18 


7.64 


35.15 


7.79 


35.11 


7.95 


36 


37 


36.19 


7.69 


36.16 


7.85 


36.12 


8.01 


36.09 


8.17 


37 


38 


37.17 


7.90 


37.13 


8.06 


37.10 


8.22 


37.06 


8.39 


38 


39 


38.15 


8.11 


38.11 


8.27 


38.08 


8.44 


38.04 


8.61 39 1 


40 
41 


39.13 
40.10 


8.32 


39.09 


8.49 
8.70 


39.05 


8.66 

8.87 


39.01 

1 


8.83 40 1 
' 9 05 'i\ 


8.52 


40.07 


40.03 


139.99 


42 41.08 


8.73 


41.04 


8.91 


41.00 


9.09 


140.96 


9.27 1 iU 


43 ■ 42.06 


8.94 


4?. 02 


9.12 


41.98 


9.3i 


141.94 


a. 49 ; 43 


44 43.04 


9.15 


43.00 


9.34 


42.96 


9.52 


42.92 9.71 44 1 


45 


44.02 


9.36 


43.98 


9.55 


43.93 


9.74 


45 89 9.93 4i 


46 


44.99 


9.56 


44.95 


9.76 


44.91 


9.96 


44.87) 10.15 1 4fi 


47 


45.97 


9.77 


45.93 


9.97 


45.89 


10.17 


45.84 1 iO.37 1 47 


48 46.95 


9.98 


46.91 


10.18 


48.86 


10.39 


46.82 


10-59, 48 


48 


47.93 


10.19 


47.88 


10.40 


47.84 


10.61 


47.79 


10 81 i 49 . 


60 


48.91 


10.40 


48.86 


10.61 


48.81 
Dep. 


10.82 


48.77 
Dep. 


11.08 1 50 1 


I 


Dep. 


1 Lat. 


Dep. 


Lat. 


Lat. 


Lat. 


1 » 
• c: 


78 


Deg, 


771 Deg. 


771 


Deg. 


77i 


Deg. 


SO 



TRAVERSE TABLE. 



97 



? 

5: 


12 Dsg. 


m Deg. 


12i Dog. 


12J Deg. 


B 
o 
a 

~6l 


Lat. 
49.89 


Dep. 

10.60 


Lat. 
49.84 


Dep. 


Lat. Dep. 


Lat. 


Dep. 


10.82 


49.79 


11.04 


49.74 


11.26 


62 


50.86 


10.81 


50.82 


11.03 


50.77 


11.25 


.50.72 


11.48 


62 


63 


61.84 


11.02 


61.79 


11.25 


51.74 


11.47 


51.69 


11.70 


63 


64 


52.82 


11.23 


52.77 


11.46 


52.72 


11.69 


52.67 


11.92 


54 


66 


63.80 


11.44 


53.75 


11.67 


53 70 i 11.90 


53.64 


12.14 


55 


66 


54.78 


11.64 


54.72 


11.88 


54.67 


12.12 


54.62 


12.36 


56 


57 


55.75 


11.85 


55.70 


12.09 


55.65 


12.34 


55.59 


12.58 


57 


58 


56.73 


12.06 


56.68 


12.31 


56.63 


12.55 


56.57 


12.80 


58 


59 


57.71 


12.27 


57.66 


12.52 


57.60 


12.77 


57.55 


13.02 


59 


60 
61 


58.69 


12.47 


58.63 


12.73 


58.58 


12.99 


58.52 


13.24 


60 
'61 


59.67 


12.68 


59.61 


12.94 


59.55 


13.20 


59.50 


13.46 


62 


60.65 


12.89 


60.59 


13.16 


60.53 


13.42 


60.47 


13.68 


62 


63 


61.62 


13.10 


61.57 


13.37 


61.51 


13.64 


61.45 


13.90 


63 


64 


62.60 


13.31 


62.54 


13.58 


62.48 


13.85 


62.42 


14.12 


64 


66 


63.58 


13.51 


63.52 


13.79 


63.46 


14.07 


63.40 


14.35 


65 


66 


64.56 


13.72 


64.50 


14.00 


64.44 


14.29 


64.37 


14.. 57 


66 


67 


65,54 


13.93 


65.47 


14.22 


65.41 


14.50 


65.35 


14.79 


67 


68 


66.51 


14.14 


66.45 


14.43 


66.39 


14.72 


66.32 


15.01 


68 


69 


67.49 


14.35 


67.43 


14.64 


67.36 


14.93 


67.30 


15.23 


69 


70 

71 


68.47 


14.55 


68.41 


14.85 


68.34 


15.15 


68.27 


15.45 


70 

71 


09.45 


14.76 


69.38 


15.06 


69.32 


15.37 


69.25 


16.67 


72 


70.43 


14.97 


70.36 


15.28 


70.29 


15.58 


70.22 


15.89 


72 


73 


71.40 


15.18 


71.34 


15.49 


71.27 


15.80 


71.20 


16.11 


73 


74 


72.38 


15.39 


72.32 


15.70 


72.25 


16.02 


72.18 


16.33 


74 


75 


73.36 


15.59 


73.29 


15.91 


73.22 


16.23 


73.15 


16.55 


75 


76 


74.34 


15.80 


74.27 


16.13 


74.20 


16.45 


74.13 


16.77 


76 


77 


75.32 


16.01 


75.25 


16.34 


75.17 


16.67 


75.10 


16.99 


77 


78 


76.30 


16.22 


76.22 


16.55 


76.15 


16.88 


76.08 


17.21 


78 


79 


77.27 


16.43 


77.20 


16.76 


77.13 


17.10 


77.05 


17.44 


79 


80 
'81 


78.25 


16.63 


78.18 
79.16 


16.97 


78.10 


17.32 


78.03 


17.66 


80 
81 


79.23 


16.84 


17.19 


79.08 


17.. 53 


79.00 


17.88 


82 


80.21 


17.05 


80.13 


17.40 


80.06 


17.75 


79.98 


18.10 


82 


83 


81.19 


17.26 


8i.ll 


17.61 


81.03 


17.96 


80.95 


18.32 


83 


84 


82.16 


17.46 


82.09 


17.82 


82.01 


18,18 


81.93 


18.54 


84 


85 


83.14 


17.67 


83.06 


IS 04 


82.99 


18.40 


82.90 


18.76 


85 


86 


84.12 


17.88 


84.04 


18.25 


83.96 


18.61 


83.88 


18.98 


86 


87 


85.10 


18.09 


85.02 


18.46 


84.94 


18.83 


84.85 


19.20 


87 


88 


86.08 


18.30 


86.00 


18.67 


85.91 19.05 


85.83 


19.42 


88 


89 


87.06 


18.50 


86.97 


18.88 


86.89 IS. 26 


86.81 


19.64 


89 


90 
91 


88.03 


18.71 


87.95 


19.10 


87.87 


19.48 


87.78 


19.86 


90 


89.01 


18.92 


88.93 


19.31 


88.84 


19.70 


86.76 


20.08 


91 


92 


89.99 


19.13 


89.91 


19.52 


89.82 


19.91 


89.72 


20.30 92 


93 


90.97 


19.34 


90.88 


19.73 


90.80 


20.13 


90.71 


20.52 93 


94 


91. 9e 


19.54 


91.86 


19.94 


91.77 


20.35 


91.68 


20.75 94 


95 


92.92 


19.75 


92.84 20.16! 


92.75 


20.56 


92.66 


20.97 95 


96 


93.90 19.96 


93.81 


20.37 


93.72 20.78 1 


93.63 


21.19 96 


97 


94.88 20.17 


94.79 


20.58 


94.70 


20.99 


94,61 


21.41 


97 


98 


95.86 20.38 


95.77 


20.79 


95.68 


21.21 


95.58 


21.63 


98 


99 


96.84 


20.58 


96.75 


21.01 


96.65 


21.43 


96.56 


21.85 


99 


100 

• 

§ 

c 

M 


97.81 


20.79 
Lat 


97.72 


21.22 


97.63 


21.64 


97.53 


22.07 


100 

9 

a 

a 

S 

CO 


Dep. 


Dep. 


Lat. 


Dep. Lat. 


Dep. 


Lat. 


78 Deg. 


771 Deg 


77 1 Deg. 


77i Deg. 



98 



trav>:rs£ table. 



1 

c 

S' 
o 

? 

T! 


13 Deg. 


1 
134 Deg. 


ISA Deg. 


13} Deg. 


C 


Lat. 1 


Dep. 1 
0.23 


Lat. ! 


Dep. ! 


Lat. 


Dep. 1 


Lat. 1 Dep. 


1 

t 


0.97 


0.97 


0.23 


0.97 0.23 1 


0.97 0.24! 


2 1 1.95 


0.45 1 


1.95 ! 


0.46 


1.95 0.47 


1.94 0.48- 


2 


J i 2 92 


0.67 


2.92 1 


0.69 


2.92 1 0.70 


2.91 0.711 


3 


4 3.90 


0.90 


3.89 


0.92 


3.89 


0.93 


3.89 0.95 


4 


fi 


4.87 


1.12 


4.87 1 


1.15 


4.86 


1.17 


4.86 1.19 


5 


fi 


5.85 


1.35 


5.84; 


1.38 


5.83 


1.40 


5.83 1.43: 


6 


7 


6.82 


].57 


6.81 


1.60 


6.81 


1.63 


6.80 1.66 


7 


8 


7.80 


1.80 


7.79 ! 


1.83 


7.78 


1.87 i 


7.77 1.90 j 


8 


9 


8.77 


2.02 


8.76 


2.06 ; 


8.75 


2.10 ; 


8.74 2.14! 


9 


10 1 

11 


9.74 
10.72 


2.25 
2.47 


9.73 


2.29 j 


9.72 2.33 


9.71 2.38: 


10 
11 


10.71 


2.52 j 


10.70 


2.57 1 


10.68 2.61 


12 


11.69 


2.70 


11.68 


2.75 


11.67 


2.80 i 


11.66 2.85 


12 


13 


12.67 


2.92 


12.65 


2.98 1 


12.64 


3.03; 


12.63 3.091 


13 


141 


13.64 


3.15 


13.63 , 


3.21 ' 


13.61 


3.27 


13.60, 3.33 


14 


1.5 


14. &i 


3.37 


14.60 


3.44 


14.59 


3.50; 


14.57 3.57 


15 


16| 


15.59 


3.60 


15.57 


3.67 


15.56 


3.74 


15.54, 3.80; 


16 


17 


16.57 


3.82 


16.55 


3.90 


16.53 


3.97 


16.51' 4.04' 


17 


18 


17.54 


4.06 


17.52 


4.13 , 


17.50 4.20 


17.48 4.28 


18 


19 


18.51 


4.27 


18.49 


4.35 


18.48 4.44 


18.46 4.52 


19 


20' 19.49 
21 20.46 


4.50 
4.72 


19.47 


4. 58 


19.45, 4.67 


19.43 


4.75 1 
4.99 ; 


20 
21 


20.44 


4.81 


20.42 1 4.90 


20.40 


22 21.44 


4.95 


21.41 


5.04 


21.39! 5.14 


21.37 5.23 


22 


23 22.41 


5.17 


22.39 


5.27 


22.36: 5.37 


22.34 5.47 


23 


24 23.38 


5.40 


23.36 


5.50 


23.34 5.50 


23.31 5.70 


24 


25 24.36 


5.62 


24.33 


5.73 


24.31 5.84 


24.28 5.94 


25 


26 25.33 


5.85 


25.31 


5.96 


125.28 6.07 


25.25 6.18 


26 


27 ; 26.31 


6.07 


26.28 


6.19 


26.25 6.30 


26.23; 6.42 


27 


28' 27.28 


6.30 


27.25 


6.42 


.27.23 6.54 


27.20 6.66 


28 


29 28.26 


6.52 


28.23 


6.65 


'28.20 6.77 
129.17 7.00 


28.17 1 6.89 


29 


30 29.23 


6.75 


1 29.20 
; 30.17 


6.88 
7.11 


29.14' 7.13 
30.11, 7.37 


30 
31 


31 1 30.21 


6.97 


'30.14 


7.24 


32 31.18 


7.20 


31.15 


7.33 


131.12 


7.47 


31.08! 7.61 


32 


33 32.15 


7.42 


32.12 


7.56 


32.09 


7.70 


32.05, 7.84 


33 


34 33.13 


7.65 


! 33.09 


7.79 


, 33.06 


7.94 


33.03 8.08 


34 


35 34.10 


7.87 


, 34.07 


8.02 


1 34.03 


8.17 


34.00 8.32 


35 


36 35.08 


: 8.10 


1 35.04 


8.25 


35.01 
35.98 


8.40 


34.97 8.56 


36 


37 36.05 


8.32 


i 36.02 


8.48 


8.64 


35.94 1 8.79 


37 


38 37.03 


8.55 


36.99 


8.71 


36.95 


8.87 


36.91 9.03 


38 


39 38.00 


8.77 


37.96 


8.94 


37.92 


9.10 


37.88 9.27 


39 


40 38.97 


9.00 


3S.94 


9.17 


1 38.89 


9.34 


38.85 9.51 


40 


41 


39.95 


9.22 


39.91 


, 9.40 


39.87 


9.57 


39.83 9.75 


41 


42 


40.92 


D.45 


40.88 


; 9.63 


40.84 


9.80 


40.80 9.98 


42 


43 


41.90 


9 67 


41.86 


9. 86 


41.81 


10.04 


41.77 10.22 


43 


44 


42.87 


9.90 


42.83 


10.03 


42.7^ 


10.27 


42.74 10.46 


, 44 


45 


' 43.85 


.0.12 


43.80 


10.31 


4«.76 


10.51 


43.71 ; 10.70 


i 45 


46 4-1.82 


10.36 


44.78 


,10.54 


44.73 


10.74 


44.68 10.93 


46 


47 


45.80 


10.57 


45.75 


, 10.77 


45.70 10.97 


45.65 11.17 


47 


48 


46 77 


10.80 


'46.72 


; 11.00 


46.67 11.21 


46.62 11.41 


48 


49 


47.74 


11.02 


47.70 


11.23 


47.65 11.44 


47.60 11.65 


49 


50 


48.72 


11.25 


4S.67 


11.46 


148.62 


11.67 


48.57 11.88 


j ^ 

8 

' S 

1 


1 
2 


' Dep. 


! Lat. 


Dep. 


1 Lat 


Dep. 


Lat. 


Dep. 


Lat. 


1 "^^ 

i 


Deg. 


76} Deg. 


1 76-5 Deg. 


76} Deg. 



rRAVERSE TABLU- 



99 



c 

5* 

P 

61 


13 Dcg 


m Deg. 


13A Deg. 


131 Deg. 


o 
a 


Lat. 


Dep. 


Lat. 


Dep. 

11.69 


Lat. 
49.59 


Dep. 


Lat. 
49.64 


Dep. 


49.69 


11.47 


49.64 


11.91 


12.12 


51 


62 


50.67 


11.70 


50.62 


11.92 


60.56 


12.14 


60.61 


12.36 ' 52 


53 


51.64 


11.92 


51.59 


12.16 


51.54 


12.37 


61.48 


12.60 , 53 


54 


62.62 


12.15 


52.66 


12.38 


62.51 


12.61 


62.45 


12.84 64 
13. 07 , 55 


55 


53.59 


12.37 


53.54 


12.61 


63.48 


12.84 


53.42 


56 


54.56 


12.60 


54.51 


12.84 


54.45 


13.07 


64.40 


1.3.31 1 56 


67 


55.54 


12.82 


55.48 


13.06 


56.43 


13.31 


56.37 


13.55^ 67 


58 


56.51 


13.05 


56.46 


13.29 


66.40 


13.54 


66.34 


13.79! 58 


59 


57.49 


13.27 


57.43 


13.52 


67.37 


13.77 


67.31 


14 ,02 


59 

60 

"6T 


60 
61 


58.46 


13.. 50 


58.40 


13.76 


58.34 
59.31 


14.01 
14.24 


68.28 
59.25 


14,26 
"14.50 


59.44 


13.72 


69.38 


13.98 


62 


60.41 


13.95 


60.35 


14.21 


60.29 


14.47 


00.22 


14.74 


62 


63 


61.39 


14.17 


61.32 


14.44 


61.26 


14.71 


61.19 


14.97 


63 


64 


62.36 


14.40 


62.30 


14.67 


62.23 


14.94 


62.17 


15.2 


64 


65 


63.33 


14.62 


63.27 


14.90 


63.20 


15.17 


63.14 


16.45 


65 


66 


64.31 


14.85 


64.24 


16.13 


64.18 


15.41 


64.11 


16.69 


66 


67 


65.28 


15.07 


65.22 


16.36 


66.15 


16.64 


65.08 


16.93 67 


68 


66.26 


15.30 


66.19 


16.69 


66.12 15.87 1 


66.05 


16.16 68 


69 


67.23 


15.52 


67.10 


16.81 


67.09 


16.11 


67.02 


16.40 69 


70 
71 


68.21 
69.18 


15.75 
15.97 


68.14 


16.04 


68.07 


16.34 


67.99 
68.97 


16.64* 
16.88 


70 

71 


69.11 


16.27 


69.04 


16.67 


72 


70.15 


16.20 


70.08 


16.50 


70.01 


16.81 


69.94 


17.11 


72 


73 


71.13 


16.42 


71.06 


16.73 


70.98 


17.04 


70.91 


17.36 ; 73 


74 


72.10 


16.65 


72.03 


16.96 


71.96 


17.28 


71.88 


17.. 59' 74 


76 


73.08 


16.87 


73.00 


17.19 


72.93 


17.60 


72.85 


17.8gl 75 


76 


74.05 


17.10 


73.98 


17.42 


73.90 


17.74 


73.82 


18.06 


76 


77 


75.03 


17.32 


74.96 


17.66 


74.87 


17.98 


74.79 


18.30 


77 


78 


76.00 


17.55 


76.92 


17.88 


75.84 


18.21 


75.76 


18.54 


78 


79 


76.98 


17.77 


76.90 


18.11 


76.82 


18.4-1 


76.74 


18.78 


79 


80 
81 


77.95 
78.92 


18.00 


77.87 


18.34 


77.79 


18.68 


77.71 


19.01 


8(, 
81 


18.22 


78.84 


18.67 


78.76 


18.91 


78.68 


19.25 


83 


79.90 


18.45 


79.82 


18.79 


79.73 


19.14 


79.65 


19.49 


82 


83 


80.87 


18.67 


80.79 


19.02 


80.71 


19.38 


80.62 


19.73 


83 


84 


81.85 


18.90 


81.76 


19.26 


81.68 


19.61 


81.59 


19.97 


84 


85 


82.82 


19.12 


82.74 


19.48 


82.65 


19.84 


82.66 


20.20 


85 


86 


83.80 


19.. S5 


83.71 


19.71 


83.62 


20.08 


83.64 


20.44 


86 


87 


84.77 


19.67 


84.68 


19.94 


84.60 


20.31 


84.51 


20.68 


87 


88 


85.74 


19.80 


85.66 


20.17 


86.57 


20.54 


85.48 


20.92 


88 


89 


86.72 


20.02 


86.63 


20.40 


86.64 


20.78 


86.45 


21.15 


89 


90 
91 


87.69 


20.25 


87.60 
88.68 


20.63 
20.86 


87.51 


21.01 


87.42 
88.39 


21.39 


90 


88.67 


20.47 


88.49 


21.24 


21.63 


91 


92 


89.64 


20.70 


89.66 


21.09 


89.46 


21.48 


89.36 


21.87 


92 


93 


90.62 


20.92 


90.62 


2^ 32 1 


90.43 


21.71 


90.33 


22.10 


93 


94 


91.59 


21.15 


91.60 


21.54 1 


91.40 


21.94 


91.31 


22.34 94 1 


95 


92.57 


21.37 


92.47 


21.77! 


92.38 


22.18 


92.28 


22.58 


95 


96 


93.54 


21.60 


93.44 


22.00 ' 


93.35 


22.41 


93.25 


22.82 


96 


97 


94 51 


21.82 


94.42 


22.23 i 


94.32 


22.64 


94.22 


23.06 


97 


98 


95 49 


22.05 


96.39 


22.46 ! 


96.29 


22.88 


95.19 


23.29, 98 1 


99 


96 46 


22.27 


96.36 


22.69 


96.26 


23.11 


96.16 


23.63 


99 


00 

e 
u 

c 
a 

■ 


97.44 


32.50 


97.34 


22.92 


97.24 


23.34 


97.13 


23 77 


100 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


c 

d 

o 
i 


7?] 


Deg. 


761 Dog. 


76:1 Dog. 


76i Deg. 



o? 



100 



TRAVERSE TABLE. 



5 

5* 

o 
o 
a 

~1 


14Deg. 


14i Deg. 


14i] 


Deg. 


141 Deg. 


5' 

a 

? 

1 


Lat. 


Dep. 
~072T 


Lat. 
0.97 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.97 


0.25 


0.97 


0.25 


0.97 


0.25 


2 


1.94 


0.48 


1.94 


0.49 


1.94 


0.50 


1.93 


0.51 


2 


d 


2.91 


0.73 


2.91 


0.74 


2.90 


0.75 


2.90 


0.76 


3 


4 


3.88, 


0.97 


3.88 


0.98 


3.87 


1.00 


3.87 


1.02 


4 


6 


4.851 


1.21 


4.85 


1.23 


4.84 


1.25 


4.84 


1.27 


5 


6 


5.82 


1.45 


5.82 


1.48 


5.81 


1.50 


5.80 


1.53 


6 


7 6.79 


1.69 


6.78 


1 72 


6.78 


1.75 


6.77 


1 78 


7 


8 


7.76 


1.94 


7.75 


1.97 


7.75 


2.00 


7.74 


2.04 


8 


9 


8.73 


2.18 


8.72 


2.22 


8.71 


2.25 


8.70 


2.29 


9 


10 

11 


9.70 
10.67 


2.42 


9.69 


2.46 


9.68 


2.50 


9.67 


2.55 

2.80 


10 
11 


2.66 


10.66 


2.71 


10.65 


2.75 


10.64 


12 


11.64 


2.90 


11.63 


2.95 


11.62 


3.00 


11.60 


3.06 


12 


13 


12.61 


3.15 


12.60 


3.20 


12.59 


3.25 


12.57 


3.31 


13 


14 


13.58 


3.39 


13.57 


3.45 


13.55 


3.51 


13.54 


3.56 


14 


15 


14.55 


3.63 


14.54 


3.69 


14.52 


3.76 


14.51 


3.82 


15 


16 


15.52 


3.87 


15.51 


3.94 


15.49 


4.01 


15.47 


4.07 


16 


17 


16.50 


4.11 


16.48 


4.18 


16.46 


4.26 


16.44 


4.33 


17 


18 


17.47 


4.35 


17.45 


4.43 


17.43 


4.51 


17.41 


4.58 


18 


19 


18.44 


4.60 


18.42 


4.68 


18.39 


4.76 


18.37 


4.84 


19 


20 

21 


19.41 


4.84 


19.38 


4.92 


19.36 


5.01 


19.34 


5.09 


20 
21 


20.38 


5.08 


20.35 


5.17 


20.33 


5.26 


20.31 


5.35 


22 


•21.35 


5.32 


21.32 


5.42 


21.30 


5.51 


21.28 


5.60 


22 


23 


22.32 


5.56 


22.29 


5.68 


22.27 


5.76 


22.24 


5.86 


23 


24 


23. S9 


5.81 


5.91 


23.24 


6.01 


23.21 


6.11 


24 


25 


24.26 


6.05 


24.23 


6.15 


24.20 


6.26 


24.18 


6.37 


25 


26 


25.23 


6.29 


25.20 


6.40 


25.17 


6.51 


25.14 


6.62 


26 


27 


26.20 


6.53 


26.17 


6.65 


26.14 


6.76 


26.11 


6.87 


27 


28 


27.17 


6.77 


27.14 


6.89 


37.11 


7.01 


27.08 


7.13 


28 


29 


28.14 


7.02 


28.11 


7.14 


28.08 


7.26 


28.04 


7.38 


29 


30 
31 


29.11 
30o08 


7.26 


29.08 


7.38 


29.04 


7.51 


29.01 


7.64 

7.89 


30 

31 


7.50 


30.05 


7.63 


30.01 


7.76 


29.98 


32 


31.05 


7.74 


31.02 


7.88 


30,98 


8.01 


30.95 


8.15 


32 


33 


32.02 


7.98 


31.98 


8.12 


31.95 


8.26 


31.91 


8.40 


33 


34 


32.99 


8.23 


32.95 


8.37 


32.92 


8.51 


32.88 


8.60 


34 


35 


33.96 


8.47 


33.92 


8.62 


33.89 


8.76 


33.85 


8.91 


35 


36 


34.93 


8.71 


34.89 


8.86 


34.85 


9.01 


34.81 


9.17 


36 


37 


36.90 


8.95 


35.86 


9.11 


35.82 


9.26 


35.78 


9.42 


37 


38 


36.87 


9.19 


36.83 


9.35 


36.79 


9.51 


36.75 


9.67 


38 


39 


37.84 


, 9.44 


37.80 


9.60 


37.76 


9.76 


37.71 


9.93 


39 


40 
41 


38.81 


1 9.68 


38.77 


9.85 


38.73 


10.02 
10.27 


38.68 


10.18 


40 

il 


39.78 


9.92 


39.74 


10.09 


39.69 


39.65 


10.44 


42 


40.75 


10.16 


40.71 


10.34 


40.66 


10.52 


40.62 


10.69 


42 


43 41.72 


! 10.40 


41.68 


10.58 


41.63 


10.77 


41.58 


10.95 


43 


44 42.69 


! 10.64 


42.65 


10.83 


42.60 


11.02 


42.55 


11.20 


44 


45 43.66 


10.89 


43.62 


11.08 


43.57 


11.27 


43.52 


11.46 


45 


46 44.63 


11.13 


44.58 


11.32 


44.53 


11.52 


44.48 


11.71 


46 


47 45.60 


11.37 


45.55 


11.57 


45.50 


11.77 


45.45 


11 97 


47 


48 1 46.57 


11.61 


46.52 


11.82 


46.47 


12.02 


46.42 


12.22 


48 


49 147.54 


11.85 


47.49 


12.06 


47.44 


12.27 


47.39 


12 48 


49 


60 ,48.51 


12.10 


48.46 


12.31 


48.41 


12.52 


48.35 


12 73 


50 

id 


i 


1 Dep 


Lat. 


Dep. 

751 


Lat. 
Deg. 


Dep. 


Lat. 


Dep. 


Lat. 

1 




76 


Deg. 


76i 


Deg. 


75i Deg. 


V 




. 








V, 





thaverse table. 



101 



5 

s 

p 
51 


14 Dog. 


U\ Deg. 


U\ Deg. 


L 


14| Deg. 


C 

a' 

g 

.1 

51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


at. 


Dep. 


49.49 


12 


.34 


49 


.43 


12.55 


49 


.38 


12.77 


49 


.32 


12.98 


62 


50.46 


12 


58 


50 


.40 


12.80 


50 


.34 


13.02 


50 


.29 


13.24 


63 


53 


51.43 


12 


82 


51 


.37 


13.05 


51 


.31 


13.27 


51 


.25 


13.49 


53 


54 


52.40 


13 


06 


52 


.34 


13.29 


52 


.28 


13.52 


52 


.22 


13.75 


54 


55 


53.37 


13 


31 


53 


.31 


13.54 


53 


.25 


13.77 


53 


.19 


14.00 


55 


56 


54.34 


13 


55 


54 


28 


13.78 


54 


.22 


14.02 


54 


.15 


14.26 


56 


6" 


55.31 


13 


79 


55 


25 


14.03 


55 


18 


14.27 


55 


.12 


14.51 


57 


5S 


56.28 


14 


03 


56 


.22 


14.28 


56 


.15 


14.52 


56 


.09 


14.77 


58 


59 


57.25 


14 


27 


57 


18 


14.52 


57 


.12 


14.77 


57 


.06 


15.02 


59 


60 
61 


58.22 


14 


52 


58 


.15 


14.77 


58 


.09 


15.02 


58 


.02 


15.28 


60 
6] 


59.19 


14 


76 


59 


.12 


15.02 


59 


06 


15.27 


58 


.99 


15.. 53 


62 


60.16 


15 


00 


60 


09 


15.26 


60 


03 


15.52 


59 


.96 


15.79 


62 


63 


61.13 


15 


24 


61 


06 


15.51 


60 


99 


15.77 


60 


.92 


16.04 


63 


64 


62.10 


15 


48 


62 


03 


15.75 


61 


96 


16.02 


61 


.89 


16.29 


64 


65 


63.07 


15 


72 


63 


00 


16.00 


62 


93 


16.27 


62 


.86 


16.55 


65 


66 


64.04 


15 


97 


63 


97 


16.25 


63 


90 


16.53 


63 


.83 


16.80 


66 


67 


65.01 


16 


21 


64 


94 


16.49 


64 


87 


16.78 


64 


79 


17.06 


67 


68 


65.98 


16 


45 


65 


91 


16.74 


65 


83 


17.03 


65 


76 


17.31 


68 


69 


66.95 


16 


69 


66 


88 


16.98 


66 


80 


17.28 


66 


73 


17.57 


69 


70 

71 


67.92 


16 


93 


67 


85 


17.23 


67 


77 


17.53 


67 


69 


17.82 


70 

71 


68.89 


17 


18 


68 


82 


17.48 


68 


74 


17.78 


68 


66 


18.08 


72 


69.86 


17 


42 


69 


78 


17.72 


69 


71 


18.03 


69 


63 


18.33 


72 


73 


70.83 


17 


66 


70 


75 


17.97 


70 


67 


18.28 


70 


59 


18.59 


73 


74 


71.80 


17 


90 


71 


72 


18.22 


71 


64 


18.53 


71 


56 


18.84 


74 


75 


72 77 


18. 


14 


72 


69 


18.46 


72 


61 


18.78 


72 


53 


19.10 


75 


76 


73.74 


18 


39 


73 


66 


18.71 


73 


58 


19.03 


73 


50 


19.35 


76 


77 


74.71 


18 


63 


74 


63 


18.95 


74 


55 


19.28 


74 


46 


19.60 


77 


78 


75.68 


18 


87 


75 


60 


19.20 


75 


52 


19.53 


75 


43 


19.80 


78 


79 


76.65 


19 


11 


76 


57 


19.45 


76 


48 


19.78 


76 


40 


20.11 


79 


80 
81 


77.62 
78.59 


19 


35 


77 


54 


19.69 


77 


45 


20.03 


77 


36 


20.37 


80 
81 


19 


60 


78 


51 


19.94 


78 


42 


20.28 


78 


33 


20.62 


82 


79.56 


19 


84 


79 


48 


20.18 


79 


39 


20.53 


79 


30 


20.88 


82 


83 


80.53 


20 


08 


80 


.45 


20.43 


80 


36 


20.78 


80 


26 


21.13 


83 


84 


81.50 


20 


32 


81 


42 


20.68 


81 


32 


21.03 


81 


23 


21.39 


84 


85 


82.48 


20 


56 


82 


.38 20.92 


82 


29 


21,28 


82 


20 


21.64 


85 


86 


83.45 


20 


81 


83 


.35 21.17 


83 


26 


21.53 


83 


17 


21.90 


86 


87 


84.42 


21 


05 


84 


.32 


21.42 


84 


23 


21.78 


84 


13 


22.15 


87 


88 


85.39 


21 


29 


85 


29 


21.66 


95 


20 


22.03 


85 


10 


22.41 


88 


89 


86.36 


21 


5ft 


86 


.26 


21.91 


86 


17 


22.28 


86 


07 


22.66 


89 


90 
91 


87.33 


21 


77 


87 


.23 


22.15 


87 


.13 


22.53 


87 


03 


22.91 


90 
91 


88.30 


22 


01 


88 


.20 


22.40 


88 


10 


22.78 


88 


00 


23.17 


92 


89.27 


22 


26 


89 


.17 


22.65 


89 


07 


23.04 


88 


97 


23.42 


92 


93 90.24 


22 


50 


90 


.14 


22.89 


90 


.04 


23.29 


89 


94 


23.68 


93 


94 91 21 


22 


.74 


91 


.11 


23.14 


91 


01 


23.54 


90 


90 


23.93 


94 


95 92 18 


22 


.98 


92 


.08 


23.38 


91 


97 


23.79 


91 


87 


24.19 


95 


9L 93 15 


23 


22 


93 


.05 


23.63 


92 


94 


24.04 


92 


84 


24.44 


96 


97 94. 12 


23 


.47 


94 


.02 


23.88 


93 


91 


24.29 


93 


80 


24.70 


97 


98 95.09 


23 


.71 


94 


.98 


24.12 


94 


88 


24.54 


94 


77 


24.95 


98 


99 96.06 


23 


95 


95 


.95 


24.37 


95 


85 


24.79 


95 


74 


25.21 


99 


IOC 


87.03 


24 


.19 


96 


.92 


24.62 


96 


.81 


25.04 


96 


.70 


25.46 


iOO 

» 

c 
a 


TO 


Dep. 


L 


at. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


76 1 


>eg. 


75| Dog 


1 


75i Deg. 


76iDeg 



102 



TRAVERSE TABLE. 



s 

g 

p 

1 


15 Deg. 


15i Deg. 


15^ Deg. 


151 Deg. 


3 

s 

1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.97 


0.26 


0.90 


0.26 


0.96 


0.27 


0.96 


0.27 


2 


1.93 


0.52 


1.93 


0.53 


1.93 


0.53 


1.92 


0.54 


2 


3| 2.90 


0.78 


2.89 


0.79 


2.89 


0.80 


2.89 


0.81 


a 


4 


3.86 


1.04 


3.86 


1.05 


3.85 


1.07 


3.85 


1.09 


4 


5 


4.83 


1.29 


4.82 


1.32 


4.82 


1.34 


4.81 


1.36 


5 


6 


5.80 


1.55 


5.79 


1.58 


5.78 


1.60 


5.77 


1.63 


6 


7 


0.76 


1.81 


6.75 


1.84 


6.75 


1.87 


6.74 


1.90 


7 


8 7 73 


2.07 


7.72 


2.10 


7.71 


2.14 


7.70 


2.17 


8 


9 


8.69 


2.33 


8.68 


2.37 


8.67 


2.41 


8.66 


2.44 


9 


10 
11 


9.66 


2.59 


9.65 


2.63 


9.64 


2.67 


9.62 


2.71 


10 


10.63 


2.85 


10.61 


2.89 


10.60 


2.94 


10.59 


2.99 


11 


12 


11.59 


3.11 


11.58 


3.16 


11.56 


3.21 


11.55 


3.26 


12 


13 


12.56 


3.36 


12.54 


3.42 


12.53 


3.47 


12.51 


3.53 


13 


14 


13.52 


3.62 


13.51 


3.68 


13.49 


3.74 


13.47 


3.80 


14 


15 


14.49 


3.88 


14.47 


3.95 


14.45 


4.01 


14.44 


4.07 


15 


16 


15.45 


4.14 


15.44 


4.21 


15.42 


4.28 


15.40 


4.34 


16 


17 


16.42 


4.40 


16.40 


4.47 


16.38 


4.54 


16.36 


4.61 


17 


18 


17.39 


4.66 


17.37 


4.73 


17.35 


4.81 


17.32 


4.89 


18 


19 


18.35 


4.92 


18.33 


5.00 


18.31 


5.08 


18.29 


5.16 


19 


20 
21 


19.32 


5.18 


19. 3C 


5.26 


19.27 


5.34 


19.25 


5.43 


20 


20.28 


5.44 


20.26 


5.52 


20.24 


5.61 


20.21 


5.70 


21 


22 


21.25 


5.69 


21.23 


5.79 


21.20 


5.88 


21.17 


6.97 


22 


23 


22.22 


5.95 


22.19 


6.05 


22.16 


6.15 


22.14 


6.24 


23 


24 


23.18 


6.21 


23.15 


6.31 


23.13 


6.41 


23.10 


6.51 


24 


25 


24.15 


6.47 


24.12 


6.58 


24.09 


6.68 


24.06 


6.79 


25 


26 


25.11 


6.73 


25.08 


6.84 


25.05 


6.95 


25.02 


7.06 


26 


27 


26.08 


6.99 


26.05 


7.10 


26.02 


7.22 


25.99 


7.33 


27 


28 


27.05 


7.25 


27.01 


7.36 


26.98 


7.48 


26.95 


7.60 


28 


29 


28.01 


7.51 


27.98 


7.63 


27.95 


7.75 


27.91 


7.87 


29 


30 
31 


28.98 


7.76 


28.94 


7.89 


28.91 


8.02 


28.87 


8.14 


30 
31 


29.94 


8.02 


29.91 


8.15 


29.87 


8.28 


29.84 


8.41 


32 


30.91 


8.28 


30.87 


8.42 


30.84 


8.55 


30.80 


8.69 


32 


33 


31.88 


8.54 


31.84 


8.68 


31.80 


8.82 


31.76 


8.96 


33 


34 


32.84 


8.80 


32.80 


8.94 


32.76 


9.09 


32.72 


9.23 


34 


35 


33.81 


9.06 


33.77 


9.21 


33.73 


9.35 


33.69 


9.50 


35 


36 


34.77 


9.32 


34.73 


9.47 


34.69 


9.62 


34.65 


9.77 


36 


37 


35.74 


9.58 


35.70 


9.73 


35.65 


9.89 


35.61 


10.04 


37 


38 


36.71 


9.84 


36.66 


10.00 


36.62 


10.16 


36.57 


10.31 


38 


39 


37.67 


10.09 


37.63 


10.26 


37.58 


10.42 


37.54 


10.59 


39 


40 
41 


38.64 


10.35 


38.59 


10.52 


38.55 


10.69 


38.50 


10.86 


to 

41 


39.60 


10.61 


39.56 


10.78 


39.51 


10.96 


39.40 


11.13 


42 40.57 


10.87 


40.52 


11.05 


40.47 


11.22 


40.42 


11.40 


42 


43 


41.53 


11.13 


41.49 


11.31 


41.44 


11.49 


41.39 


11.67 


43 


44 


42.50 


11.39 


42.45 


11.57 


42.40 


11.76 


42.35 


11.94 


U 


45 


43.47 


11.65 


43.42 


11.84 


43.36 


12.03 


43.81 


12.21 


45 


16 


44.43 


11.91 


44.38 


12.10 


44.33 


12.29 


44.27 


12.49 


4G 


47 45.40 


12.16 


45 35 


12.36 


45.29 


12.56 


45.24 


12 76 


47 


48 46.36 


12.42 


46.31 


12.63 


46.25 


12.83 


46.20 


13.03 


48 


49 47.33 


12.68 


47.27 


12.89 


47.22 


13.09 


47.16 


13.30 


49 


50 


,48.30 


12.94 


48.24 


13.15 


48.18 


13.36 


48.12 


13.. 57 


50 

§ 

^ 


s 

1 

CD 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 


75 


Dog. 


741 


Deg. 


U^ 


Deg. 


• 744 Deg. 



TRAVERSE TABLE. 



103 



D 

a 
a 
a 

61 


15 Dcg. 


15i Deg 


15A Deg. 


16J Deg. 


1 

o 

p 

'61 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 

13.84 


49.26 


13.20 


49.20 


13.41 


49.15 


13.63 


49.09 


62 


50.23 


13.46 


50.17 


13.68 


50.11 


13.90 


.50.05 


U.U 


52 


63 


61.19 


13.72 


51.13 


13.94 


51.07 


14.16 


51.01 


14,39 


53 


64 


62.16 


13.98 


52.10 


14.20 


52.04 


14.43 


51.97 


14.60 


54 


55 


53.13 


14.24 


53.06 


14.47 


53.00 


14.70 


.52.94 


14.93 


5£ 


66 


54.09 


14.49 


54.03 


14.73 


53.96 


14.97 


53.90 


15.20 


56 


57 


55.06 


14.75 


54.99 


14.99 


54.93 


15.23 


54.86 


15.47 


67 


58 


56.02 


15.01 


55.96 


15.26 


55.89 


15.50 


55.82 


15.74 


58 


59 


56.99 


15.27 


56.92 


15.52 


56.85 


15.77 


56.78 


16.01 


59 


60 
61 


57.96 


15.53 


57.89 


15.78 


57.82 
58.78 


16.03 


57.75 


16.29 
16.56 


60 
61 


58.92 


15.79 


58.85 


16.04 


16.30 


58.7] 


62 


59.89 


16.05 


59.82 


16.31 


59.75 


16.57 


59.67 


16.83 


62 


63 


60.85 


16.31 


60.78 


16.57 


60.71 


16.84 


60.63 


17.10 


63 


64 


61.82 


16.56 


61.75 


16.83 


61.67 


17.10 


61.60 


17.37 


64 


65 


62.79 


16.82 


62.71 


17.10 


62.64 


17.37 


62.56 


17.64 


65 


66 


63.75 


17.08 


63.68 


17.36 


63.60 


17.64 


63.52 


17.92 


66 


67 


64.72 


17.34 


64.64 


17.62 


64.56 


17.90 


64.48 


18.19 


67 


68 


65.68 


17.60 


65.61 


17.89 


65.53 


18.17 


65.45 


18.46 


68 


69 


66.65 


17.86 


66.57 


18.15 


66.49 


18.44 


66.41 


18.73 


69 


70 
71 


67.61 


18.12 


87.54 


18.41 


67.45 


18.71 


67.37 


19.00 


70 

71 


68.58 


18.38 


68.50 


18.68 


68.42 


18.97 


68.33 


19.27 


72 


69.55 


18.63 


69.46 


18.94 


69.38 


19.24 


69.30 


19.54 


72 


73 


70.51 


18.89 


70.43 


19.20 


70.35 


19.51 


70.26 


19.82 


73 


74 


71.48 


19.15 


71.39 


19.46 


71.31 


19.78 


71.22 


20.09 


74 


75 


72.44 


19.41 


72.36 


19.73 


72.27 


20.04 


72.18 


20.86 


75 


76 


73.41 


19.67 


73.32 


)9.99 


73.24 


20.31 


73.15 


20.63 


76 


77 


74.38 


19.93 


74.29 


20.25 


74.20 


20.58 


74.11 


20.90 


77 


78 


75.34 


20.19 


75.25 


20.52 


75.16 


20.84 


75.07 


21.17 


78 


79 


76.31 


20.45 


76.22 


20.78 


76.13 


21.11 


76.03 


21.44 


79 


80 
81 


77.27 


20.71 


77.18 


21.04 


77.09 
78.05 


21.38 


77.00 


21.72 


80 
81 


78.24 


20.96 


78.15 


21.31 


21.65 


77.96 


21.99 


82 


79.21 


21.22 


79.11 


21.57 


79.02 


21.91 


78.92 


22.26 


82 


83 


80.17 


21.48 


80.08 


21.83 


79.98 


22.18 


79.88 


22.53 


83 


84 


81.14 


21.74 


81.04 


22.09 


80.94 


22.45 


80.85 


22.80 


84 


85 


82.10 


22.00 


82.01 


22.36 


8]. 91 


22.72 


81.81 


23.07 


85 


86 


83.07 


22.26 


82.97 


22.62 


82.87 


22.98 


82.77 


23.34 


86 


87 


84.04 


22.52 


83.94 


22.88 


83.84 


23.25 


83.73 


23 62 


87 


88 


85.00 


22.78 


84.90 


23.15 


84.80 


23.52 


84.70 


23 89 


88 


89 


85.97 


23.03 


85.87 


23.41 


85.76 


23.78 


85.66 


24 16 


89 


90 
91 


86.93 


23.29 


86.83 


23.67 


86.73 


24.05 


86.62 


24 43 
24.70 


90 
"91 


87.90 


23.55 


87.80 


23.94 


87.69 


24.32 


87.58 


92 


88.87 


23.81 


88.76 


24.20 


88.65 


24.59 


88.55 


24.97 


92 


93 


89.83 


24.07 


89.73 


24.46 


89.62 


24.85 


89.51 


25.24 


93 


94 


90.80 


24.33 


90.69 


24.72 


90.. 58 25.12 


90.47 


25.52 


94 


95 


91.76 24.59 


91.65 


24.99 


91 34 25.39 


91.43 


25.79 


95 


96 


92 73 24.85 


92.62 


25.25 


92.51 i 25.65 


92.40 


26.06 


96 


97 


93.69 25.11 


93.58 


25.51 


93.47 


25.92 


93.36 


26.33 


97 


98 


94.66 


25.36 


94., 55 


25.78 


94.44 


26.19 


91:. 32 


26.60 


98 


99 


95.63 


25.62 


95.51 


26.04 


95.40 


26.46 


95.28 


26.87 


99 


100 

§ 

e 

d 

.a 


96.59 


25.88 
Lat. 


1 36.48 


26.30 


96.36 


26.72 
Lat. 


96.25 


27.14 


100 

1 

4-) 


Dep. 


Dep. Lat. 


Dep. 


Dep. 


Lat. 


75 Deg. 

1 


741 Dcg. 


74^ Deg. 


74i Deg. 



104 



TRAV£U8£ TABLE. 



1 

Ol 
5 1 

r* 
P 


16 Deg. 


1 
I6i Deg. 


\^\ 


Deg. 


i 
16| Deg, 1 

1 


s 
\ 

1 


1 
1 


Lat. 

0.96 


Dep. 


Lat. Dep. 


Lat. 


D©p. 

28 


Lat. 


Dep. i 


0.28 


0.96 


0.28 


0.96 


0.96 


0.29 


2 


1 9'^ 55 


1.92 


0.56 


1.92 


0.57 


1.92 


0.58 


2 


3 


2.88 C 83 


2.88 


0.84 


2.88 


0.85 


2.87 


0.86 


3 


4 


3.85] 


1.10 


3.84 


1.12 


3.84 


1.14 


3.83 


1.15 4| 


6 


4.81 


1.38 


4.80 


1.40 


4.79 


1.42 


4.79 


1.44 


5 


6 


5.77 


1.65 


5.76 


1.68 


5.75 


1.70 


5.75 


1.73 


6 


7 


6.73 


1.93 


0.72 


1.96 


6.71 


1.99 


6.70 


2.02 


7 


8 


7.69 


2.21 


7.68 


2.24 


7.67 


2.27 


7.66 


2.31 


8 


9 


8.65 


2.48 


8.64 


2.52 


8.63 


2.56 


8.62 


2.59 


9 


10 
11 


9.01 
10.. 57 


2.76 


9.60 


2.80 


9.59 


2.84 


9.58 


2.88 


10 
11 


3.03 


10.56 


3.08 


10.55 


3.12 


10.53 


3.171 


12 


11.64 


3.31 


11.52 


3.36 


11.51 


3.41 


11.49 


3.46 


12 


13 


12.50 


3.58 


12.48 


3.64 


12.46 


3.69 


12.45 


3.75 


13 


14 


13.46 


3.86 


13.44 


3.92 


13.42 


3.98 


13.41 


4.03 


14 


15 


14.42 


4.13 


14.40 


4.20 


14.38 


4.26 


14.36 


4.32 


15 


16 


15.38 


4.41 


15.36 


4.48 


15.34 


4.54 


15.32 


4.61 


16 


17 


16.34 


4.69 


16.32 


4.76 


16.30 


4.83 


16.28 


4.90 


17 


18 


17.30 


4.961 


17.28 


5.04 


17.26 


5.11 


17.24 


5.19 


18 


19 


18.26 


5.24 


18.24 


5.32 


18.22 


5.40 


18.19 


5.48 


19 


20 
21 


19.23 


5.51 


19.20 


5.60 


19.18 


5.68 


19.15 


5.76 


20 
'21 


20.19 


5.79 


20.16 


5.88 


20.14 


5.96 


20.11 


6.05 


22 


21.15 


6.06 


21.12 


6.16 


21.09 


6.25 


21.07 


6.34 


22 


23 


22.11 


6.34 


22.08 


6.44 


22.05 


6.53 ! 


22.02 


6.63 


23 


24 


23.07 


6.62 


23.04 


6.72 


23.01 


6.82 1 


22.98 


6.92 


24 


25 


24.03 


6.89 


24.00 


7.00 


23.97 


7.10 


23.94 


7.20 


25 


20 


24.99 


7.17 


24.96 


7.28 


24.93 


7.38 


24.90 


7.49 


26 


27 


25.95 


7.44 


25.92 


7.56 


25.89 


7.67 


25.85 


7.78 


27 


28 


26.92 


7.72 


20.88 


7.84 


26.85 


7.95 


26.81 


8.07 


28 


29 


27.88 


7.99 


27.84 


8.11 


27.81 


8.24 


27.77 


8.36 


29 


30 
31 


28.84 
29.80 


8.27 


28.80 


8.39 


28.76 


8.52 


1 28 . 73 
29.68 


8.65 
8.93 


30 
31 


8.54 


29.76 


8.67 


29.72 


8.80 


32 


30.76 


8.82 


.30.72 


8.95 


30.68 


9.09 


30.64 


9.22 


32 


33 


31.72 


9.10 


31.68 


9.23 


31.64 


9.37 


31.60 


9.51 


33 


34 


32.68 


9.37 


32.64 


9.51 


32.60 


9.66 


32.56 


9.80 


34 


35 


33.64 


9.65 


33.60 


9.79 


33.56 


9.94 


33.51 


10.09 


35 


36 


34.61 


9.92 


34.56 


10.07 


34.52 


10.22 


34.47 


10.38 


36 


37 


35.57 


10.20 


35 52 


10.35 


35.48 


10.51 


35.43 


10.66 


37 


38 


36.53 


10.47 


36.48 


10.63 


36.44 


10.79 


36.39 


10.95 


38 


39 


37.49 


10.75 


37.44 


10.91 


37.39 


11.08 


37.35 


11.24 


39 


40 
41 


38.45 


11.03 


38.40 


11.19 
11.47 


38.35 


11.36 


38.30 


11.53 


' 40 
41 


39.41 


11.30 


39.36 


39.31 


11.64 


39.26 


11.82 


42 


40.37 


11.58 


40.32 


11.75 


40.27 


11.93 


40.22 


12.10 


42 


43 


41.33 


11.85 


41.28 


12.03 


41 23 


12.21 


41.18 


12.39 


43 


44 


43.30 


12.13 


42 . 24 


12.31 


42. 19 


12.50 


42.13 


12.68 1 44 


45 


43.26 


12.40 


43.20 


12.59 


43. 15 


12.78 


43.09 


12.97 45 


46 


44.22 


12.68 


44.16 


12.87 


44.11 


13.06 


44.05 


13.26 


46 


47 


45.18 


12.95 


45.12 


13.15 


45.00 


13.35 


45.01 


18.55 


47 


48 


46.14 


13.23 


46.08 


13.43 


46.02 


13.63 


45.96 


13.83 


48 


49 


47.10 


13.51 


47.04 


"».71 


46.98 


13.92 


46.92 


14.12 


49 


60 

.3 


48.06 


13.78 
Lat. 


48.00 


13.99 


47.94 


14.20 
Lat. 


47.88 
Dep. 


14.41 


50 

o 

{ i 

i s 


Dep. 


Dep. 


Lat. 


Dep. 


LaU 


74 Deg. 


73| 


Deg. 


73^ Deg. 


734 Dog. 



TKA.VKnSl! TABLE. 



105 



f 

51 


16 Deg. 


16i Deg. 


16A Deg 


m Deg. 












r* 


Lat. 

49,02 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 

48.84 


Dep. 
14.70 


O 

o 
51 


14.06 


48.96 


14.27 


48.90 


14.48 


52 


49.99 


14.33 


49.92 


14.55 


49 86 1 


14.77 


49.79 


14.99 


52 


53 


50.95 


14.61 


.50.88 


14 83 


.50.82 1 15.05 1 


50.75 


15.27 


53 


54 


51.91 


14.88 


51.84 


15 11 


51.78 


15. .34 1 


51.71 


15.66 


64 


55 


52.87 


15.16 


.52.80 


15.39 


.52.74 


15.62 


.52.67 


16.85 


55 


66 


,'S3.93 


15.44 


.53.76 


15.67 


.53.69 


15.90 


.53.62 


16.14 


56 


57 


64.79 


15.71 


54 . 72 


15.95 


.54.65 


16.19 


.54.58 


16.43 


57 


58 


55.75 


15.99 


55.68 


16.23 


.55.61 


16.47 


55.. 54 


16.72 


58 


69 


56.71 


16.26 


.56.64 


16.51 


.56 . 57 


16.76 


56.. 50 


17.00 


59 


60 
"61 


57.68 
58.64 


16.54 


.57.60 


16.70 


57.. 53 
.58.49 


17.04 
17.32 


57.45 

58.41 


17.29 
17.58 


60 
'61 


16.81 


.58.56 


17.07 


62 


59.60 


17.09 


59 . 52 


17.35 


.59.45 


17.61 


.59.37 


17.87 


62 


63 


60.56 


17.37 


50 48 


17.63 


60.41 


17.89 


60.33 


18.16 


63 


64 


61.52 


17.64 


61.44 


17.91 


61.36 


18.18 


61.28 


18.44 


04 


65 


62.48 


17.92 


62.40 


18.19 


62.32 


18.46 


62 . 24 


18.73 


65 


66 


63.44 


18.19 


63.30 


18.47 


63.28 


18.74 


63.20 


19.02 


60 


67 


64.40 


18.47 


64.32 


18.75 


64.24 


19.03 


64.16 


19.31 


67 


68 


65.37 


18.74 


65.28 


19.03 


65.20 


19.31 


65 . 1 1 


19.60 


68 


69 


66.33 


19.02 


66.24 


19.31 


66.16 


19.60 


66.07 


19.89 


69 


70 
71 


67.29 1 


19.29 
19.57 


67.20 


19.59 


67.12 
68.08 


19.88 


67.03 


20.17 


70 
71 


68.25 


68.16 


19.87 


20 . 1 7 


67.99 


20.46 


72 


69.21 


19.85 


69.12 


20.15 


69.03 


20.45 


68.95 


20.75 


72 


73 


70.17 


20.12 


70.08 


20.43 


69.99 


20 . 73 


69.90 


21.04 


73 


74 


71.13 


20.40 


71.04 


20.71 


70.95 


21 02 


70.86 


21.33 


74 


75 


72.09 


20.67 


72.00 


20.99 


71.91 


21.30 


71.82 


21.61 


75 


76 


73.06 


20.95 


72.96 


21.27 


72.87 


21.. 59 


72.78 


21.90 


76 


77 


74.02 


21.22 


73.92 


21.. 55 


73.83 


21.87 


73.73 


22.19 


77 


78 


74.98 


21. .50 


74.88 


21.83 


74.79 


22.15 


74 . 69 


22.48 


78 


79 


75.94 


21.78 


75.84 


22.11 


75 . 75 


22.44 


75.65 


22.77 


79 


80 
81 


76.90 


22.05 


76.80 


22.39 


76.71 


22.72 
23.01 


1 76.61 
S77..56 


23.06 


80 
81 


77.86 


22.33 


77.76 


22.67 


77.66 


23.34 


82 


78.82 


22.60 


78.72 


22.95 


78 . 62 


23.29 


78 . 52 


23.63 


82 


83 


79.78 


22.88 


79.68 


23.23 


79.. 58 


23 . 57 


'79.48 


23.92 


83 


84 


80.75 


23.15 


80.64 


23..^ I 


80 . .54 


23.86 


80.44 


24.21 


84 


85 


81.71 


23.43 


81.60 


23 . 79 


81.50 


24.14 


81. .39 


24.. 50 


85 


86 


82.67 


23.70 


82.56 


24.07 


82.46 


24.43 


82.35 


24.78 


86 


87 


83.63 


23.98 


83.. 52 


24.35 


83.42 


24.71 


j 83.31 


25.07 


87 


88 


84.59 


24.26 


84.48 


24.62 


84.38 


24.99 


84.27 


25.36 


88 


89 


85.55 


24.53 


85.44 


24.90 


85.33 


25 , 28 


85.22 


25.65 


89 


90 
91 


86.51 


24.81 


86.40 


25.18 


86.29 


25.. 56 


86.18 
87.14 


25.94 


90 


87.47 


25.08 


87.. 36 


25.46 


87.25 


25.85 


26.23 


^ 


92 


88.44 


25.36 


88.32 


25.74 


88.21 


26.13 


88.10 


26.51 


92 


93 


89.40 


25.63 


89.28 


26.02 


89.17 


26.41 


89.05 


26.80 


y3 


94 


90.36 


25.91 


90.24 


26.30 


90.13 


26.70 


90.01 


27.09 


94 


95 


91.32 


26.19 


91.20 


26.. 58 


91.09 


26.98 


90.97 


27.38 


95 


96 


92.28 


26.46 


92.16 


26.86 


92.05 


27.27 


91.93 


27.67 


96 


97 


93.24 


26.74 


93.12 


27.14 


93.01 


27.. 55 


92.88 


27.95 


97 


98 


94.20 


27.01 


94.08 


27.42 


^93.96 


27.83 


93.84 


28.24 


98 


99 


95.16 


27.29 


95.04 


27.70 


94.92 


28.12 


94.80 


28.. 53 


99 


100 


96.13 


27.56 


96.00 


27.98 


95.88 


28.40 


95.76 


28.82 


100 

i 

c 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 










(d 


.9 


I 74 Deg. 


73} Deg. 


^^ Deg. 


7'H Deg. 


o 



106 



THAVERSE TABLB. 



5 

CB 

f 

3 


17 Deg. 


17i Deg. 


n^Deg. i 

1 


I7i Deg. 


y 

s 

S 
8, 


Lat. 


D«p. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. ' 

0.30 


1 


0.96 


0.29 


0.95 


0.30 


0.95 


0.30 


0.95 


*i 


1.91 


0.58 


1.91 


0.59 


1.91 


0.60 


1.90 


0.61 


a 


3 


2.87 


0.88 


2.87 


0.89 


2.86 


0.90; 


2.86 


0.91 


9 


4 


3.83 


1.17 


3.82 


1.19 


3.81 


1.20 


3.81 


1.22 


4 


5 


4.78 


1.46 


4.78 


1.48 


4.77 


1.50 


4.76 


1.52 


6 


6 


5.74 


1.75 


5.73 


1.78 


5.72 


1.80 


5.71 


1.83 


6 


7 


6.69 


2.05 


6.69 


2.08 


6.68 


2.10! 


6.67 


2 13 


7 


8 


7.65 


2.34 


7.64 


2.37 


7.63 


2.41 


7.62 


2.44 


8 


9' 8.61 


2.63 


8.60 


2.67 


8.58 


2.71 i 


8 57 


2.74 


9 


10 9.56 


2.92 


9.55 


2.97) 


9.54 


3.01 


9.. 52 


3.05 


10 
11 


11 


10.52 


3.22 


10.51 


3.26 


10.49 


3.31 


10.48 


3.35 


la 


11.48 


3.51 


11.46 


3.56 1 


11.44 


3.61 


11.43 


3.66 


12 


13 


12.43 


3.80 


12.42 


3.85 i 


12.40 


3.91 


12.38 


3.96 


13 


14 


13.39 


4.09 


13.37 


4.15 i 


13.35 


4.21 1 


13.33 


4.27 


14 


15 


14,34 


4.39 


14.33 


4.45! 


14.31 


4.51 ! 


14.29 


4.57 


15 


16 


15.30 


4.68 


15.28 


4.74 i 


15.26 


4.81 


15.24 


4.88 


16 


17 


16.26 


4.971 


16.24 


5.04 


16.21 


5.11 


16.19 


5.18 


17 


18 


17.21 


5.26 


17.19 


5.34 


17.17 


5.41 


17.14 


5.49 


18 


19 


18.17 


5.56 


18.15 


5 63 


18.12 


5.71 


18.10 


5.79 


19 


20 
21 


19.13 


5.85 


19.10 


5.93 


19.07 


6.01 


19.05 


6.10 


20 

21 


20.08 


6.14 


20.06 


6. S3 


20.03 


6.31 


20.00 


6.40 


22 


21.04 


6.43 


21.01 


6 52 


20.98 


6.62 


20.95 


6.71 


22 


23 


21.99 


6.72 


21.97 


6.82 


21.94 


6.92; 


21.91 


7.01 


23 


24 


22.95 


7.02 


2:i.92 


7.12; 


22.89 


7.22 


22 . 86 


7.32 


24 


2.^ 


23.91 


7.31 


23.88 


7 41 


23.84 


7.. 52 


23.81 


7.62 


25 


26 


24.86 


7.60 


24.83 


7.71 


24.80 


7.82 


24.76 


7.93 


26 


27 


25.82 


7.89 


25.79 


8.01, 


25.75 


8.12 


25.71 


8.23 


27 


28 


26.78 


8.19 


26.74 


8.30 


26-70 


8.42 


26.67 


8.54 


28 


29 


27.73 


8.48 


27.70 


8.60 


27.66 


8.72 


27.62 


8.84 


29 


30 
31 


28 . 69 


8.77 ; 28.65 
9.06 129.61 


8.90 


28.61 
29.57 


9.02 


28.57 


9.15 


JO 
31 


29.35 


9.19 


9.32 


29.52 


9.45 


32 


30.60 


9.36 i! 30.56 


9.49 


30.52 


9.62' 


30.48 


9.76 


32 


33 


31.56 


9.65 1 31.. 52 


9.79 1 


31.47 


9.92 


31.43 


10.06 


33 


34 


32.51 


9.94 32.47 


ia.08 1 


32.43 


10.22 


32.08 


10.37 


34 


35 


33.47 


10.23 33.43 


10.38 1 


33.. 38 


10.52 


33.33 


10.67 


35 


36 


34.43 


10.53 1 .34.. 38 


10.68! 


34.33 


10.83 


34.29 


10.98 


36 


37 


35.38 


10.82 35.34 


10.97] 


35.29 


11.13 


35 24 


11.28 


37 


38 


36.34 


11.11 36.29 1 11.271 


36.24 


11.43 


36.19 


11.58 


38 


39 


37.30 


11.40 37.25 


11.57 1 


37.19 


1 1 . 73 


37.14 


11.89 


39 


40 
41 


as. 25 


11.09 ; 33.20 


11.86 


38.15 


12.03 


38.10 


12.19 


40 

41 


39.21 


11.99 39.16 


12.16 


39.10 


12.33 


39.05 


12.50 


42 


40.16 


12. 2S 40.11 


12.45 


40.06 


12.63 


40.00 


12.80 


42 


43 


41.12 


12.57 41.07 


12.75 


41.01 


12.93 


40.95 


13 11 


43 


44 42. OS 


12.86 42.02 


13.05 


41.96 


13.23 


41.91 


13 11 


44 


45 143. 3 


13.16 42.98 


13.34 


42.92 


13.53 


42.86 


13 72 


45 


46 ! 43.99 


13.45 43.93 


13.64 


43.87 


13.83 


43.81 


14.02 


46 


47 44.95 


18.74 44.89 


13.94 


44.82 


14.13 


44.76 


14.. 33 


47 


48 ,45.90 


14.03 45.84 


14.23 


45.78 


14.43 


45.71 


14 63 


48 


49 ) 46.86 


14.33 46.80 


14.53 


46.73 


14.73 


46.67 


14.94 


, 49 


50 147.82 
i Dep. 

a 73 


14.62 
Lat. 

Deg. 


47.75 


14.83 


47.69 


15.04 


47.62 


15.24 


50 

a 


1 Dep. 


Lat. 


Dep. 


Lat 


Dep. 

72i 


Lat. 
Der 


721 


Deg. 


791 


Deg. 



TRAVEBSB TABLS. 



107 



g 
s 

a 
? 

5l 


17 Deg. 


m Deg, 


17A Deg. 


1?| Deg. 


O 

S 

p 
"51 


Lat. 
48.77 


Dep. 
14.91 


Lat. 


Dep. 
15.12 


Lat. 


Dep. 


Lat. 


Dop. 


48.71 


48.64 


15.34 


48.67 


16.55 


62 


49.73 


15.20 


49.66 


15.42 


49.59 


15.64 


49.52 15.85 


52 


53 


50.68 


15.50 


50.62 


15.72 


50.55 


15.94 


50.48 


16.16 


53 


54 


51.64 


15.79 


51.57 


16.01 


51.50 


16.24 


51.43 


16.46 


54 


55 


52.60 


16.08 


52.53 


16.31 


52.45 


16.54 


52.38 


16.77 


55 


56 


53.55 


16.37 


53.48 


16.61 


53.41 


16.84 


53.33 17.07 


56 


67 


54.51 


16.67 


54.44 


16.90 


.54.36 


17.14 


54.29 


17.38 


57 


58 


55.47 


16.96 


55.39 


17.20 


55.32 


17.44 


55.24 


17.68 


58 


59 


56.42 


17.25 


56.35 


17.50 


56.27 


17.74 


56.10 


17.99 


59 


60 
61 


57.38 


17.54 


57.30 


17.79 


57.22 


18.04 


57.14 


18.29 


60 
61 


58.33 


17.83 


58.26 


18.09 


58.18 


18.34 


58.10 


18.60 


62 


59.29 


18.13 


59.21 


18.39 


59.13 


18.64 


.59.05 


18.90 


62 


63 


60.25 


18.42 


60.17 


18.68 


60.08 


18.94 


60.00 


19.21 


63 


64 


61.20 


18.71 


61.12 


18.98 


61.04 


19.25 


60.95 


19.51 


64 


65 


62.16 


19.00 


62.08 


19.28 


61.99 


19.55 


61.91 


19.82 


65 


66 


63.12 


19.30 


63.03 


19.57 


62.95 


19.35 


62.86 


20.12 


66 


67 


64.07 


19.59 


63.99 


19.87 


63.90 


20.15 


63.81 


20.43 


67 


68 


65.03 


19.88 


64.94 


20.16 


64.85 


20.45 


64.76 


20.73 


68 


69 


65.99 


20.17 


65.90 


20.46 


65.81 


20.75 


65.72 


21.04 


69 


70 
71 


66.94 


20.47 


66.85 


20.76 


66.76 


21.05 


66.67 


21.34 


70 
"71 


67.90 


20.76, 


67.81 


21.05 


67.71 


21.35 


67.62 


21.65 


72 


68.85 


21.05! 


68.76 


21.35 


68.67 


21.65 


68.57 


21.95 


72 


73 


69.81 


21.34 1 


69.72 


21.65 


69.62 


21.95 


69.52 


22.26 


73 


74 


70.77 


21.64 


70.67 


21.94 


70.58 


22.25 


70.48 


22.56 


74 


75 


71.72 


21.93 


71.63 


22.24 


71.53 


22.55 


71.43 


22.86 


75 


76 


72.68 


22.22 


72.. 58 


22.54 


72.48 


22.85 


72.38 


23.17 


76 


77 


73.64 


22.51 


73.54 


22.83 


73.44 


23.15 


73.33 


23.47 


77 


78 


74.59 


22.80 


74.49 


23.13 


74.39 


23.46 


74.29 


23.78 


78 


79 


75.55 


23.10 


75.45 


23.43 


75.34 


23.76 


75.24 


24.08 


79 


80 
81 


76.50 
77.46 


23.39 


76.40 


23.72 


76.30 


24.06 


76.19 


24.39 


80 


23.68 


77.36 


24.02 { 


77.25 


24.36 


77.14 


24.69 


81 


82 


78.42 


23.97 


78.31 


24.32 


78.20 


24.66 


78.10 


25.00 


82 


83 


79.37 


24.27 


79.27 


24.61 


79.16 


25.96 


79.05 


25.30 


83 


84 


80.33 


24.56 


80.22 


24.91 \ 


80.11 


25.26 


80.00 


25.61 


84 


85 


81.29 


24.85 


81.18 


25.21 


81.07 


25.56 


80.95 


25.91 


85 


86 


82.24 


25.14 


82.13 


25.50 


82.02 


25.86 


81.91 


26.22 


86 


87 


83.20 


25.44 


83.09 


25.80 


82.97 


26.16 


82.86 


26.52 


87 


88 


84.15 


25.73 


84.04 


26.10 


83.93 


26.46 


183.81 


26.83 


88 


89 


85.11 


26.02 


85.00 


26.39 


84.88 


26.76 


84.76 


27.13 


89 


90 
91 


86.07 
87.02 


26.31 


85.95 


26.69 185.83 


27.06 
27.36 


85.72 


27.44 


90 

91 


26.61 


86.91 


26.99 


86.79 


86.67 


27.74 


92 


87 98 


26.90 


87.86 


27.28 


87. ?4 


27.66 


87.62 


28.05, 92 1 


93 


88.94 


27.19 


88.82 


27.58 


88.70 127.97 


88.57 


2?. 35 


93' 


94 


89.89 


27.48 


89 . 77 


27.87 


89.65 '28.27 


89.53 


28.66 


94 


95 


90.85 


27.78 


90.73 


28.17 


90.60 


■28., 57 


90.48 


28.96 


95 


96 


91 82 


28.07 


91.68 


28.47 


91.56 


28.87 


91.43 


29.27 


96 


97 


92. 7< 28.36 


92.64 


28.76 


92.51 


29.17 


92.38 ! 29.5? 


97 


98 


93.72 


28.65 


93.59 


29.06 


93.46 


129.47 


93.33 29.88 


98 


99:94.67 


28.94 


94.55 


29.36 


94.42 


29.77 


94.29 30.18 


99 


100 

S 

e 
• 

1 


95.63 


29.24 


^5.50 


29.65 
Lat. 


95.37 


30 07 


95.24 30.49 


100 

s 

a 
a 

CO 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 


Dep. Lat. 


73 Deg. 


72i 


Deg. 


72^ Dog. 


7Si Deg. 



108 



THAVERSE TAULE. 



5 

e 

P 

1 


' 18 Deg. 


1 m Deg. 


1 

l^ Deg. 


I8| Deg. 


m 

s 


' Lai. 
"0.95 


Dep. 


Lat- i Dep. 

1 
' 0.95 ' 0.31 


Lai. 

. 0,95 


Dep. 


Lau 


1 ^^P- 


0.31 


0.32 


0.95 


0.32. 1 


2 


1.90 


0.62 


1.90 


0.63 


1 1.90 


0.63 


' 1.89 


9.64 3 


3 1 2.85 


0.93 


2.85 


3.94 


2 84 


0.95 


' 2.84 


0.96 3 


4 


3.80 1.24 


, 3.80 


1.25 


3 79 


1.27, 


3.79 


1 29 4 


5 


4.76 1 1.55 


4.75 


1.57 


4.74 


1.59 


4.73 1.61 6 


6 


5.711 1.85 


5.70 


1.88 


5.69 


1.90 


,5.68 1.93 6 


7 


6.66 2.16 


6.65 


2.19 


6.64 


2.22 


1 6.63 


2.25 7 


8 


7.61 2.47 


7.60 


2.51 


7.59 


2.54 


7.58 


2.57 B 


9 


8.56} 2.78 


S.55 


2.82 


1 8.. 53 


2.86 1 


1 8.52 


2.89 9 


10 
11 


9.51 3.09 
10.46. 3.40 


9.50 


3.13 


' 9.48 
■ 10.43 


3.17 
3.49! 


9.47 
10.42 


3.21 ! 10 
3.54 11 


10.45 


3.44 


12 


11.41 3.71 


11.40 


3.76 


i 11.38 


J 81 I 


' 11.36 


3.86 12 


13 


12.36 4 02 


12.35 


4.07 


12.33 


4 12 


12.31 


4.18 13 


14 


13.31 4 33 


13.30 


4.38 


; 13.28 


4.4i 13.26 


4.50 14 


15 


14.27 1 4.64 


14.25 


4.70 


14.22 


4.76 


14.20 


4.82 15 


16 


15.22 4.94 


15.20 


5.01 


: 15.17 


5.08 


15.15 


5.14 16 


17 


16.17, 5.25 


16.14 


5.32 


1 16.12 


5.39 


16.10 


5.46 17 


18 


17.12 ' 5.56: 


17.09 


5.64 


17.07 


5.71 


17.04 


5.79 18 


19 


18.07 5.87 


18.04 


5.95 


18.02 


6.03 


17.99 


6.11 19 


20 
21 


19.02^ 6.18 
19.97^ 6.49 


13 99 
19.94 


6.26 
6.58 


18.97 


6.35 


18.94 
19.89 


6.43 20 
6.75 21 


19.91 


6.66 


22 


20.92' 6.80, 


20.89 


6.89 


20.86 


6.98 


20.83 


7.07 22 


23 


21.87' 7.11! 


21.84 


7.20 


21.81 


7.30 


21.78 7.39 23 


24 


22.83' 7.42 


22.79 


7.52 


22 . 76 


7.62 


22.73 


r 71 24 


25 


23.78 ; 7.73 


23.74 


7.83' 


23.71 


7.93 


23.67 


8.04 25 


26 


24.73 8.03 


24.69 


8.14 


24,66 


8.25 


24.62 


8.36 26 


27 


25.68 , 8.34 


25.64 


8.46 


25.60 


8.57 ' 


25.57 


8.68 27 


28 


26.63 8.65 


26.59 


8.77 


26.55 


8.88, 


26.51 


9.00 28 


29 


27.58 ! 8.96 


27.54 


9.08 


27.50 


9.20 


27.46 


9.32 29 


30 

31 


28.53 9.27 
29.48 1 9.58 


28.49 


9.39 


28.45 
29.40 


9.52 
9.84 


28.41 
29.35 


9.64 3C 
9.96 31 


29.44 


9.71 


32 


30.43; 9.89. 


30.39 


10.02 


.30.35 


10.15 


30.30 


10.29 32 


33 


31.38 1 10.20 


31.34 


10.33 


31.29 


10.47 


31.25 


10.61 33 


34 


.32.34 ]0.51 


32.29 


10.65 


32 . 24 


10.79 


32.20 


10.93 34 


3.=i 


33.29 10.82 


33.24 


10.96 


33.19 


11.11 


33.14 


11.25 35 


36 


34.24 11.12 


34.19 


11.27 


.34.14 


11.42 


34.09 


11.57 36 


37 


35 19 11.43 


35.14 


11.59 


35.09 


1 1 . 74 


35.04 


11.89 37 


3S 


36.14 11.74 


36.09 


11.90 


36.04 


12.06 


35.98 


12.21 38 


39 


37.09 12.05 


37.04 


12.21 


36 . 9S 


12.37 


36.93 


12.54 .39 


40 


38.04 12.36 


37.99 


12.53 


37.93 


12.69 


37.88 
38.82 


12.86 ' 40 
13.18 ; 41 


41 


38.99 12.67 


38 . 94 


12.84 


3S.SS 


13.01 1 


42 


39.94 12.98 


39.89 


13.15 


39.83 


13.33 


39.77 


13.50 42 


43 


40.90 13.29 


40.84 


13.47 


40.78 


13 64 


40.72 


13.82 43 


44 


41.85 13.60 


41.79 


13.78 


41.73 


13.96 


41.66 


14.14 U 


it 42.80 13.91 


42.74 


14.09 


42.67 


14. 2S 


42.61 


14.46 45 


46 43.75 14.21 


43.69 


14.41 


43.62 


14.60 


43.56 


14.79 46 


47 44.70 14.. =52 


44.64 


14.72 1 


44.57 


14.91 


44.51 


15.11 47 


i8 45.65 14.83 


45.59 


15.03! 


45.52 


15.23 


45.45 


15.43 48 


49 46.60 15.14 


46.54 


15.35 


46.47 


15.55 


46.40 


15.75 49 


50 47.55 


15.45. 


47.48 


15,66 


47.42 


15.87 


47.35 


16.07 50 


8 ; Dep. 


Lat. 


Dep. ' 


Lai. 


bep. 


Lat. : 


Dep. 


Lat. 


g 


1 ' 
1 72 Deg. 


711 Deg. 


Tii 


1 
Deg. 1 


7U Deg. 




1 






1 



TRAVITRSK TABLK. 



109 



1' 

a 
a 
a 

61 


18 Dog. 


18i Deg. 


18^ Deg. 


181 Deg. 


51 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 

16.18 


Lat. 

48.29 


Dep. 
76.^9 


48.50 


15.76 


48.43 


15.97 


48.36 


52 


49.45 


16.07 


49.38 


16.28 


49.31 


16.50 


49.24 


16 71 


52 


53 


50.41 


16.38 


50.33 


16.60 


50.26 


16.82 


50.19 


17.04 63 


54 


51.36: 16.69 


51.28 


16.91 


51.21 


17.13 


51.13 


17.36 54 


55 


52.31 


17.00 


52.23 


17.22 


52.16 


17.45 


52.08 


17.68 55 


56 


53.26 


17.30 


53.18 


17.54 


53.11 


17.77 


53.03 


18.00 1 56 


57 


54.21 


17.61 


54.13 


17.85 


54.05 


18.09 


53.98 


18.32 i 57 


58 


55.16 


17.92 


55.08 


18.16 


55.00 


18.40 


54.92 


18.64 ' 58 


69 


56.11 


18.23 


5().03 


18.48 


55.95 


18.72 


55.87 


18.96 


59 


60 
61 


57.06 
58.01 


18.54 


56.98 


18.79 


56.90 


19.04 


56.82 


19.29 


60 
61 


18 85 


57.93 


19.10 


57.85 


19.36 


57.76 


19.61 


62 


58.97 


19.16 


58.88 


19.42 


58.80 


19.67 


58.71 


19.93 


62 


63 


59.92 


19.47 


59.83 


19.73 


59.74 


19.99 


59.66 


20.25 


63 


64 


60.87 


19.78 


60.78 


20.04 


60.69 


20.31 


60.60 


20.57 


64 


65 


61.82 


20.09 


61.73 


2G.36 


61.64 


20.62 


61.55 


20.89 


65 


66 


62.77 


20.40 


62.68 


20.67 


62.59 


20.94 


62.50 


21.22 


66 


67 


63.72 


20.70 


63.63 


20.98 


63.. 54 


21.26 


63.44 


21.54 


67 


68 


64.67 


21.01 


64.58 


21.30 


64.49 


21.58 


64.39 


21.86 


68 


69 


65.62 


21.32 


65.53 


21.61 


65.43 


21.89 


65.34 


22.18 


69 


70 
71 


66.57 


21.63 


66.48 


21.92 
22.23 


66.38 
67.33 


22.21 


66.29 


22.50 


70 
71 


67.53 


21.94 


67.43 


22.53 


67.23 


22.82 


72 


68.48 


22.25 


68.38 


22.55 


68.28 


22.85 


68.18 


23.14 


72 


73 


69.43 


22.56 


69.33 


22.86 


69.23 


23.16 


69.13 


23.47 


73 


74 


70.38 


22.8? 


70.28 


23.17 


70.18 


23.48 


70.07 


23.79 


74 


75 


71.38 


23.18 


71.23 


23.49 


71.12 


23.80 


71.02 


24.11 


75 


76 


72.28 


23.49 


72.18 


23.80 


72.07 


24.12 


71.97 


24.43 


76 


77 


73 23 


23.79 


73.13 


24.11 


73.02 


24.43 


72.91 


24.75 


77 


78 


74.18 


24.10 


74.08 


24.43 


73.97 


24.75 


73.86 


25.07 


78 


79 


75.13 


24.41 


75.03 


24.74 


74.92 


25.07 


74.81 


25.39 


79 


80 

81 


76.08 


24.72 


75.98 


25.05 


75.87 


25.38 


75.75 


25.72 


80 
81 


77.04 


25.03 


76.93 


25.37 


76.81 


25.70 


76.70 


26.04 


82 


77.99 


25.34 


77.88 


25.68 


77.76 


26.02 


77.65 


26.36 


82 


83 


78.94 


25.65 


78.83 


25.99 


78.71 


26.34 


78.60 


26.68 


83 


84 


79.89 


25.96 


79.77 


26.31 


79.66 


26.65 


79.54 


27.00 


84 


85 


80.84 


26.27 


80.72 


26.62 


80.61 


26.97 


80.49 


27.32 


85 


86 


81.79 


26.58 


81.67 


26.93 


81.56 


27.29 


81.44 


27.64 


86 


87 


82.74 


26.88 


82.62 


27.25 


82.50 


27.61 


82.38 


27.97 


87 


88 


83.69 


27.19 


83.57 


27.. 56 


83.45 


27.92 


83.33 


28.29 


88 


89 


84.64 


27.50 


84.52 


27.87 


84.40 


28.24 


84.28 


28.61 


89 


90 
91 


85.60 


27.81 


85.47 


28.18 


85.35 


28.56 


85.22 


28.93 


90 
91 


86.55 


28.12 


86.42 


28.50 


86.30 


28.37 


86.17 


29.25 


92 


87.50 


28.43 


87.37 


28.81 


87.25 


29.19 


87.12 


29,57 


92 


93 


88.45 


28.74 


8y.32 


29.12 


88.19 


29.51 


88.06 


29.89 


93 


94 


89 40 


29.05 


89.27 


29.44 


89.14 


29.83 


89.01 


30.22 


94 


95 


90 35 


29.36 


90.22 


29.75 


90.09 


30.14 


89.96 


30.54 


95 


96 


91.30 


29.67 


91.17 


30.06 


91.04 


30.46 


5,0.91 


30.86 


96 


97 


92.25 


29.97 


92.12 


30.38 


91.99 


30.78 


91.85 


31.18 


97 


98 


93.20 


30.28 


93.07 


30.69 


92.94 


31.10 


92.80 


31.50 


98 


9C 


94.15 


30.59 


94.02 


31.00 


93.88 


31.41 


93.75 


31.82 


99 


IOC 

g 

at 


95.11 


30.90 


94.97 


31.32 
Lat. 


94.83 


31.73 


94.69 


32 14 


.00 

§ 
a 

cd 
2 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 


Dep. 


Lat, 


72 Deg. 


711 Deg. ! 


n^ Deg. 


7U Deg. 



no 



TRAVERSE TABLE. 



5 


19 Deg. 


19i Deg. 


19i 


Deg. 


19J Dog. 




§ 


Lat. 


Dop. 
0.33 


Lat. 


Dep. 


Lat. 
0.94 


Dep. 
0.33 


Lat. 

0.94 


Dep. 
0.34 


e.96 


0.94 


0.33 


Z ..H9 


0.65 


1.89 


0.66 


1.89 


0.67 


1.88 


0.68 


2 


? 2.84 0.98 


2.83 


0.99 


2.83 


l.UO 


2 82 


1.01 


8 


4 3 . 78 . 1 30 


3.78 


1.32 


3.77 


1.34 


3.76 


1 35 


4 


5 4.73 


1.63 


4.72 


1.66 


4.71 


1.67 


4.71 1.69 


6 


6 


6.67 


1.95 


6.66 


1.98 


6.66 


2.00 


6.66 2.03 


fi 


7 


6.62 


2.28 


6.61 


2.81 


6.60 


2.34 


6.69, 2.37 


7 


8 


7.56 2.60 1 


7.65 


2.64 


7.54 


2.67 


7.63 


2.70 


S 


9 


8.51 


2.93 


8.60 


2.97 


8.48 


3.00 


8.47 


3.04 


9 


10 

11 


9.46 
10.40 


3.26 
3.58 


9.44 
10.38 


3.30 
3.63 


9.43 
10.37 


3.34 


9.41 


3.38 


10 
11 


3.67 


10.35 


3.72 


12 


11.35 


3.91 


11.33 


3.96 


11.31 


4.01 


11.29 


4.06 


12 


13 


12.29 


4.23 


12.27 


4.29 


12.25 


4.34 


12.24 


4.39 


13 


14 


13.24 


4.56 


13.22 


4.62 


13.20 


4.67 


13 18 


4.73 


14 


15 


14.18 


4.88 


14.16 


4.95 


14.14 


5.01 


14.12 


5.07 


15 


16 


15.13 


5.21 


15.11 


6.28 


15.08 


6.34 


16.06 


5.41 


16 


17 


16.07 


5.53 


16.06 


5.60 


16.02 


5.67 


16 00 


5.74 


17 


18 


17.02 


6.86 


16.99 


6.93 


16.97 


6.01 


16.94 


6.08 


18 


19 


17.96 


6.19 


17.94 


6.26 


17.91 


6.34 


17.88 


6.42 


19 


20 

21 


18.91 


6.51 


18.88 


6.69 


18.86 
19.80 


6.68 


18.82 


6.76 


20 
21 


19.86 


6.84 


19.83 


6.92 


7.01 


19.76 


7.10 


22 


20.80 


7.16 


20.77 


7.25 


20.74 


7.34' 


a0.71 1 7.43 


22 


23 


21.75 


7.49 


21.71 


7.58 


21.68 


7.68! 


21.66 1 7.77 


23 


24 


22.69 


7.81 


22.66 


7.91 


22.62 


8.01 


22.69, 8.11 


24 


25 


23.64 


8.14 


23.60 


8.24 


23.67 


8.36 


23.63 8.45 


25 


26 


24.58 


8.46 


24.66 


8.57 


24.51 


8.68 


24.47 1 8.79 


26 


27 


25 . 53 


8.79 


25.49 


8.90 


25.45 


9.01 


25.41 


9.12 


27 


28 


26.47 


9.12 


26.43 


9.23 


26 . 39 


9.35 


26.35 


9.46 


28 


29 


27.42 


9r44 


27.38 


9.56 i 


27 . 34 


9.68 


27.29 


9.80 


29 


30 


28.37 


9.77 


28.32 


9.89 


28 . 28 


10.01 


28.24 


10.14 


30 


21 


29.31 


10.09 


29.27 


10.22 


29.22 


10.35 


29. 18 


10 48 


31 


32 


30.26 


10.42 


30.21 


10.55 


30.16 


10.68 


30.12 


10 81 


32 


33 


31.20 


10.74 


31.16 


10.88 


31.11 


11.02 


31.06 


11 16 


33 


34 


32.15 


11.07 


32.10 


11.21 


32.05 


1 1 . 36 


32.00 


11 49 


34 


35 


33.09 


11.39 


33.04 


1 1 . 54 


32.99 


11.68 


32.94 


11.83 


35 


36 


34.04 


1 1 . 72 


33.99 


11.87 


33.94 


12.02 


33.88 


12.17 


36 


37 


34.98 


12.05 


34.93 


12.20 


34.88 


12.36 


34.82 


12.50 


37 


38 


35.93 


12.37 


35.88 


12.53 


35.82 


12.68 


35.76 


12.84 


38 


39 


36.88 


12.70 


36.82 


12.86 


36.76 


13.02 


36.71 


13.18 


39 


40 


37 82 


13.02 


37.76 


13.19 


37.71 


13.35 


37.65 


13.62 


40 


41 


38.77 


13.35 


38.71 


13.52 


38.05 


13.69 


38.59 


13.85 


41 


42 39 71 


13.67 


39.65 


13.86 


39.69 


14.02 


39.63 


14.19 


42 


43 40.66 


14.00 


40.60 


14. :8 


40 . 53 


14.36 


40.47 


14. 53 


43 


44 41.60 


14.32 


41.54 


14.51 


41.48 


14.69 


41.41 


14 87 


44 


45 42.55 


14.66 


42.48 


14.84 


42.42 


16.02 


42.36 


15.21 


45 


46 ,43.49 


14.98 


43.43 


15.17 


43.36 


16.36 


43.29 


16.64 


46 


47 


44.44 


16.30 


44.37 


15.50 


44.30 


16.69 


44.24 


16.88 


47 


48 


45.38 


16.63 


45.32 


15.83 


45.25 


16.02 


46. L8 


16.22 


48 


49 


46.33 


x6.96 


46.26 


16.16 


46.19 


16.36 


46.12 


16.66 


49 


50 

8 

a 


47.28 


16.28 


47-20 


16.48 
Lat. 


47.13 


16.69 
Lut. 

Deg. 


47.06 


16.90 


60 


Dcp. 


Lat. 


Dep. 


Dcp. 

70^] 


Dep. 


Lai. 


s| 


71 Deg. 


701 


Deg. 


7(H Deg. 


G • 



THA.VEK6B TABLE. 



Ill 



s 

61 


19 Deg. 


19i Deg. 


19| Deg. 


19| Deg. 


O 










? 














Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


o 
? 

51 


48.22! 16.60 1 


48.15 


16.81 


48.07 


17.02 


48.00 


17.23 


63 49.17; 16.93 | 


49.09 


17.14 


49.02 


17.30 


48.94 


17.57 


52 


63 


50 11 17.26 1 


50.04 


17.47 


49.96 


17.69 


49.88 17 91 


53 


64 


51.06 


17.58 


50.98 


17.80 


50.90 


18.03 


.50.82 18 25 


54 


55 


52.00 


17.91 


51.92 18.13 1 


51.85 


18.36 


51.76 


18.59 


55 


56 


52.95 


18 23 


52.87 


18.46 


52.79 


18.69 


52.71 


18.92 


56 


57 


53.89 


18.56 


53.81 


18.79 


53.73 


19.03 


53.65 


19.26 


57 


58 


54.84 


18.88 


54.76 


19.12 


54.67 


19.36 


54.59 


19.60 


58 


59 


55.79 


19.21 


55.70 


19.45 


55.62 


19.69 


55.53 


19.94 


59 


60 
61 


56.73 


19.53 


56.65 
57.59 


19.78 


56.56 
57.50 


20.03 


56.47 


20.27 


60 
61 


57.68 


19.86 


20.11 


20.36 


57.41 


20.61 


62 


58.62 


20.19 


58.53 


20.44 


58.44 


20.70 


58.35 


20.95 


62 


63 


59.57 


20.51 


59.48 


20.77 


59.39 


21.03 


59.29 


21.29 


63 


64 


60.51 


20.84 


60.42 


21.10 


60.33 


21.36 


60.24 


21.63 


64 


65 


61.46 


21.16 


61.37 


21.43 


61.27 


21.70 


61.18 


21.96 


65 


66 


62.40 


21.49 


62.31 


21.76 


62.21 


22.03 


02.12 


22.30 


66 


67 


63.35 


21.81 


63.25 


22.09 


63.16 


22.37 


63.06 


22.64 


67 


68 


64.30 


22.14 


64.20 


22.42 


04.10 


22.70 


64.00 


22.98 


68 


69 


65.24 


22.40 


65.14 


22.75 


65.04 


23.03 


64.94 


23.32 


69 


70 
71 


66.19 


22.79 


66.09 


23.08 


65.98 


23.37 


65.88 


23.65 


70 
71 


67.13 


23.12 


67.03 


23.41 


66.93 


23.70 


66.82 


23.99 


72 


68.08 


23.44 


67.97 


23.74 


67.87 


24.03 


67.76 


24.33 


72 


73 


69.02 


23.77 


68.92 


24.07 


68.81 


24.37 


68.71 


24.67 


73 


74 


69.97 


24.09 


69.86 


24.40 


69.76 


24.70 


69.65 


25.01 


74 


75 


70.91 


24.42 


70.81 


24.73 


70.70 


25.04 


70.59 


25.34 


75 


76 


71.86 


24.74 


71.75 


25.06 


71.64 


25.37 


71.53 


25.68 


76 


77 


72.80 


25.07 


72.69 


25.39 


72.58 


25.70 


72.47 


26.02 


77 


78 


73.75 


25.39 


73.64 


25.72 


73.53 


26.04 


73.41 


26.36 


78 


79 


74.70 


25.72 


74.58 


26.05 


74.47 


26.37 


74.35 


26.70 


79 


80 
81 


75.64 


26.05 


75.53 


26.38 


75.41 


26.70 


75.29 


27.03 


80 
81 


76.59 


26.37 


76.47 


26.70 


76.35 


27.04 


76.24 


27.37 


82 


77.53 


26.70 


77.42 


27.03 


77.30 


27.37 


77.18 


27.71 


82 


83 


78.48 


27.02 


78.36 


27.36 


78.24 


27.71 


78.12 


28.05 


83 


84 


79.42 


27.35 


79.30 


27.69 


79.18 


28.04 


79.06 


28.39 


84 


85 


80.37 


27.67 


80.25 


28.02 


80.12 


28.37 


80.00 


28.72 


85 


86 


81.31 


28.00 


81.19 


28.35 


81.07 


28.71 


80.94 


29.06 


86 


87 


82.26 


28.32 


82.14 


28.68 


82.01 


29.04 


81.88 


29.40 


87 


88 


83.21 


28.65 


83.08 


29.01 


92.95 


29.37 


82.82 


29.74 


88 


89 


84.15 


28.98 


84.02 


29.34 


83.90 


29.71 


83.76 


30.07 


89 


90 
91 


85.10 


29.30 


84.97 


29.67 


84.84 


30.04 


84.71 


30.41 


90 
31 


86.04 


29.63 


85.91 


30.00 


85.78 


30.38 


85.65 


30.75 


92 


86.99 


29.95 


86.86 


30.33 


86.72 


30.71 


86.59 


31.09 


92 


93 


87.93 


30.28 


87.80 


30.66 


87.67 


31.04 


87.. 53 


31.43 


9.9 


94 


88.88 


30.60 


88.74 


30.99 


88.61 


31.38 


88.47 


31.76 


94 


95 


89.82 


30.93 


89.69 


31.32 


89.55 


31.71 


89.41 


32.10 


95 


96 


90.77 


31.25 


90.63 


31.65 


90.49 


32.05 


90.. 35 


32.44 


96 


97 


91.72 


31.58 


91.58 


31.98 


91.44 


32.38 


91.29 


32.78 


97 


98 


92.66 


31.91 


92.52 


32.31 


92.38 


32.71 


92.24 


33.12 


98 


99 


93.61 


32.23 


93.46 


32.64 


93.32 


33.05 


93.18 


33.45 


99 


.DC 


94.65 


32.56 


94.41 


32.97 


94.26 


33.38 


94.12 


33.79 


100 

i 

Q 


s 

K 

o 

1 11 ■■■ 


Dep. 


L&t. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


«^ 


Oeg. 


701 Deg. 


70^ Deg. 


70i Deg. 



112 



TRAVERSE TABLE. 



5' 

s 

a 
? 

~1 


20] 


Deg. 


m Deg. 


20^ Deg. 


201 Deg. 


S' 

p 

' 1 


Lat. 


Dep. 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 
0,ZI 


0.94 


0.34 


0.94 


0.35 


0.94 


0.35 


0.94 


2 


1.88 


0.68 


1.88 


0.69 


1.87 


0.70 


1.87 


0.71 


i 


3 


2.82 


1.03 


2.81 


1.04 


2.81 


1.05 


2.81 


1.06 


3 


4 


3.76 


1.37 


3.75 


1.38 


3.75 


1.40 


3.74 


1.42 


i 


5 


4.70 


1.71 


4.69 


1.73 


4.68 


1.75 


4.68 


1.77 


5 


G 


5.64 


2.06 


6.63 


2.08 


5.62 


2.10 


5.6^ 


2.13 


6 


7 


6.68 


2.39 


6.67 


2.42 


6.66 


2.45 


6.55 


2 48 


7 


8 


7.52 


2.74 


7.61 


2.77 


7.49 


2.80 


7.48 


2.83 


8 


9 


8.46 


3.08 


8.44 


3.12 


8.43 


3.15 


8.42 


3.19 


9 


10 
11 


9.40 


3.42 


9.38 


3.46 


9.37 
10.30 


3.50 


9.35 


3.54 


lO 
11 


10.34 


3.76 


10.32 


3.81 


3.85 


10.29 


3.90 


12 1 11.28 


4.10 


11.26 


4.15 


11.24 


4.20 


11.22 


4.25 


12 


13 


12.22 


4.45 


12.20 


4.50 


12.18 


4.56 


12.16 


4.61 


13 


14 


13.16 


4.79 


13.13 


4.86 


13.11 


4.90 


13.09 


4.96 


14 


15 


14.10 


5.13 


14.07 


6.19 


14.06 


6.25 


14.03 


6.31 


15 


16 


16.04 


5.47 


16.01 


5.64 


14.99 


6.60 


14.96 


6.67 


16 


17 


16.97 


6.81 


15.96 


5.88 


15.92 


5.95 


15.90 


6.02 


17 


18 


16.91 


6.16 


16.89 


6.23 


16.86 


6.30 


16.83 


6.38 


18 


19 


17.85 


6.60 


17.83 


6.58 


17.80 


6.66 


17.77 


6.73 


19 


20 

21 


18.79 


6.84 


18.76 


6.92 


18.73 


7.00 ! 


18.70 


7.09 


20 
21 


19.73 


7.18 


19.70 


7.27 


19.67 


7.35 


19.64 


7.44 


22 


20.67 


7.62 


20.64 


7.61 


20.61 


7.70 


20.57 


7.79 


22 


23 


21.61 


7.87 


21.58 


7.96 


21.54 


8.05 


21.51 


8.15 


23 


24 


22.55 


8.21 


22.52 


8.31 


22.48 


8.40 


22.44 


8.50 


24 


25 


23.49 


8.56 


23.45 


8.65 


23.42 


8.76 


23.38 


8.86 


26 


26 


24.43 


9.89 


24.39 


9.00 


24.35 


9.11 


24.31 


9.21 


26 


27 


25.37 


9.23 


25.33 


9.35 


25.29 


9.46 


25.25 


9.57 


27 


28 


26.31 


9.58 


26.27 


9.69 


26.23 


9.81 


26.18 


9.92 


28 


29 


27.26 


9.92 


27.21 


10.04 


27.16 


10.16 


27.12 


10.27 


29 


30 
31 


28.19 


10.26 


28.15 


10.38 


28.10 


10.51 


28.06 


10.63 


30 
31 


29.13 


10.60 


29.08 


10.73 


29.04 


10.86 


28.99 


10.98 


32 


30.07 


10.94 


30.02 


11.08 


29.97 


11.21 


29.92 


11.34 


32 


33 


31.01 


11.29 


30.96 


11.42 


30.91 


11.66 


30.86 


11.69 


38 


34 


31.95 


11.63 


31.90 


11.77 


31.85 


11.91 


31.79 


12.06 


,34 


35 


32.89 


11.97 


32.84 


12 11 


32.78 


12.26 


32.73 


12.40 


35 


36 


33.83 


12.31 


33.77 


12.46 


33.72 


12.61 


33.66 


12.75 


36 


37 


34.77 


12.65 


34.71 


12.81 


34.66 


12.96 


34.60 


13.11 


37 


38 


35.71 


13.00 


35.66 


13.16 


36.59 


13.31 


35.54 


13.46 


38 


39 


36.65 


13.34 


36.59 


13.60 


36.53 


13.66 


36.47 


13.82 


39 


40 
41 


37.59 


13.68 
14.02 


37.53 


13.84 


37.47 


14.01 


37.41 


14.17 


40 
41 


38.53 


38.47 


14.19 


38.40 


14.36 


38.34 


14.53 


42 


39.47 


14,36 


39.40 


14.64 


39.34 


14.71 


39.28 


14.88 


42 


43 


40.41 


14.71 


40.34 


14.88 


40.28 


15.06 


40.21 


15.23 


43, 


44 


41.35 


15.06 


41.28 


16.23 


41.21 


15.41 


41.15 


15.59 


44 


45 42.29 


16.39 


42.22 


15.68 


42.15 


16.76 


42.08 


:5.94 


45 


46 143.23 


16.73 


43.16 


15.92 


43.09 


16.11 


43.02 


16 30 


46 


4- f 44.17 


16.07 


44.09 


16.27 


44.02 


16.46 


43.95 


16 65 


47 


48 45. ix 


16.42 


46.03 


16.61 


44.96 


16 81 


44.89 


17.01 


48 


49 46.04 


16.76 


45.97 


16.96 


46.90 


17.16 


45.82 


17.36 


49 


60 

i 

a 

.2 
Q 


46.98 


17.10 


46.91 


17.31 


46.83 


17.51 


46.76 


17.71 


50 

« 

a 

ed 


Dep. 


Lat. 


Dep. 

691 


Lat. 
Deg. 


Dep. 


Lat. 


Dep. 


Lat. 


701 


Oeg. 


69i Deg. 


69i 


Deg. 



TRAVERSE TABLE. 



113 



3 
? 

51 


20Deg. 


20t De?. 


20ADeg 


201 Deg. 


5 

m 

§ 

a 
51 


Lat. 


Dep. 


Lat. 


Dep. 
■T7.'65 


Lat. 

47777 


Dep. 


Lat. 


Dep. 


47.92 


17.44 


47.85 


17.86 


47.69 


18.07 


52 


48.86 


17.79 


48.79 


18.00 


48.71 


18.21 


48.63 


18.42 


52 


53 


49.80 


18.13 


49.72 


18.34 


49.64 


18.56 


49.56 


18.78 


53 


54 


50.74 


18.47 


50.66 18.69 1 


50.58 


18.91 


50.50 


19.13 


54 


55 


51.68 


18.81 


51.60 


19.04 


51.52 


19.26 


51.43 


19.49 


55 


56 


52.62 


19.15 


52.54 


19.38 


52.45 


19.61 


52.37 


19.84 


56 


57 


53.56 


19.50 


53.48 


19.73 


53.39 


19.96 


53.30 


20.19 


57 


58 


54.50 


19.84 


54.42 


20.07 


54.33 


20.31 


54.24 


20.55 


58 


59 


55.44 


20.18 


55.35 


20.42 


55.26 


20.66 


55.17 


20.90 


59 


60 

61 


56.38 


20.. 52 


56.29 


20.77 


56.20 


21.01 


56.11 


21.26 


60 
'61 


57.32 


20.86 


57.23 


21.11 


57.14 


21.36 


57.04 


21.61 


62 


58.26 


21.21 


58.17 


21.46 


58.07 


21.71 


57.98 


21.97 


62 


63 


59.20 


21.55 


59.11 


21.81 


59.01 


22.06 


58.91 


22.32 


63 


64 


60.14 


21.89 


60.04 


22.15 


59.95 


22.41 


59.85 


22.67 


64 


65 


61.08 


22.23 


60.98 


22.50 


60.88 


22.76 


60.78 


23.03 


65 


66 


62.02 


22.57 


61.92 


22.84 


61.82 


23.11 


61.72 


23.38 


66 


67 


62.96 


22.92 


62.86 


23.19 


62.76 


23.46 


62.65 


23.74 


67 


68 


63.90 


23.26 


63.80 


23.54 


63.69 


23.81 


63.59 


24.09 


68 


69 


64.84 


23.60 


64.74 


23.88 


61". 63 


24.16 


64.52 


24.45 


69 


70 
71 


65.78 


23.94 

24.28 


65.67 


24.23 


65.57 
66.50 


24.51 


65.46 


24.80 


70 

'71 

72 


66.72 


66.61 


24.57 


24.86 


66.39 


25.15 


72 


67.66 


24.63 


67.55 


24.92 


67.44 


25.21 


67.33 


25.51 


73 


68.60 


24.97 


68.49 


25.27 


68.38 


25.57 


68.26 


25.86 


73 


74 


69.54 


25.31 1 


69.43 


25.61 


69.31 


25.92 


69.20 


26.22 


74 


75 


70.48 


25.65 


70.36 


25.96 


70.25 


26.27 


70.14 


26.57 


75 


76 


71.42 


25.99 


71.30 


26.30 


71.19 


26.62 


71.07 


26 . 93 


76 


77 


72.36 


26.34 


72.24 


26.65 


72.12 


26.97 


72.01 


27.28 


77 


78 


73.30 


26.68 


73.18 


27.00 


73.06 


27.32 


72.94 


27.63 


78 


79 


74.24 


27.02 


74.12 


27.34 


74.00 


27.67 


73.88 


27.99 


79 


80 

81 


75.18 1 27.36 1 


75.06 


27.69 


74.93 

75.87 


28.02 


74.81 


28.34 


80 
81 


76.12 


27.70 


75.99 


28.04 


28.37 


75.75 


28.70 


82 


77.05 


28.05 


76.93 


28.38 


76.81 


28.72 


76.68 


29.05 


82 


.^3 


77.99 


28.39 


77.87 


28.73 


77.74 


29.07 


77.62 


29.41 


83 


84 


78.93 


28.73 


78.81 


29.07 


78.68 


29.42 


78.55 


29.76 


84 


85 


79.87 


29.07 


79.75 


29.42 


79.62 


29.77 


79.49 


30.11 


85 


86 


80.81 


29.41 


80.68 


29.77 


80.55 


30.12 


80.42 


30.47 


86 


87 


81.75 


29.76 


81.62 


30.11 


81.49 


30.47 


81.36 


30.82 


87 


88 


82.69 


30.10 


82.56 


30.46 


82.43 


30.82 


82.29 


31.18 


88 


89 


83.63 


30.44 


83.50 


30.80 


83.36 


31.17 


83.23 


31.58 


89 


90 
91 


84.57 


30.78 


84.44 


31.15 


84.30 


31.52 


84.16 


31.89 


90 
91 


85.51 j 31.12 


85.38 


31.50 


85.24 


31.87 


85.10 


32 24 


92 


86.45 1 31.47 


86.31 


31.84 


86.17 


32.22 


86.03 


32.59 


92 


93 


87.39 1 31.81 


87.25 


32.19 


87. Ix 


32.57 


86.97 


32.95 


93 


94 


88.33 32.15 


88.19 


32.54 


88.05 


32.92 


87.90 


33.30 


94 


95 


89.27: 32.49 


89.13 


32.88 


88.98 


33.27 


88.84 


33.66 


95 


96 


90.21 32.83 


90.07 


33.23 


89.92 


33.62 


89.77 


34.01 


96 


97 


91.15 33.18 


91.00 


33.57 


90.86 


33.97 


90.71 


34.37 


97 


98 


92.09 


33.52 


91.94 


33.92 


91.79 


34.32 


91.64 


34.72 


98 


99 


93.0 J 


33.86 


92.88 


34.27 


92.73 


34.67 


92.58 


35.07 


99 


£00 

1 


93.97 


34.20 


93.82 


34.61 


93.67 


35.02 


93.51 


35.43 


100 

U 

s 

3 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


70 


Deg. 


69| Deg. 


69^ De^ 


69i Ueg 



114 



TRAVERSE TABLE. 



o 


21 Deg. 


2U Deg. 


211 


Deg. 


211 Deg. 


5 




1 




















Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


9 

a 
? 


0.93 


0.33 


0.93 


0.36 


0.^3 


0.37 


0.93 


0.37 


2 


1.87 


0.72 


1.86 


0.72 


1.86 


0.73 


1.86 


0.74 


2 


8 


2.80 


1.08 


2.80 


1.09 


2.79 


1.10 


2.79 


1.11 


3 


4 


3.73 


1.43 


3.73 


1.45 


3.72 


1.47 


3.72 


1.48 


4 


6 4.67 


1.79 


4.66 


1.81 


4.65 


1.83 


4.64 


1.85 


5 


6 f 


5.60 


2.15 


5.59 


2.17 


6.58 


2.20 


5.57 


2.22 


6 


7 


6.54 


2.51 


6.52 


2.54 


6.61 


2.57 


6.50 


2.59 


7 


8 


7.47 


2.87 


7.46 


2.90 


7.44 


2.93 


7.43 


2.96 


8 


9 


8.40 


3.23 


8.39 


3.26 


8.37 


3.30 


8.36 


3.34 


9 


10 


9.34 


3.58 


9.32 


3.62 


9.30 


3.67 


9.29 


3.71 


10 

11 


11 


JO. 27 


3.94 


10.25 


3.99 


10.23 


4.03 


10.22 


4.08 


12 


11.20 


4.30 


11.18 


4.35 


11.17 


4.40 


11.15 


4.45 


12* 


13 


12.14 


4.66 


12.12 


4.71 


12.10 


4.76 


12.07 


4.82 


13 


14 


13.07 


5.02 


13.05 


5.07 


13.03 


5.13 


13.00 


5.19 


14 


15 


14.00 


5.38 


13.98 


5.44 


13.96 


5.. 50 


13.93 


5.56 


15 


16 


14.94 


5.73 


14.91 


5.80 


14.89 


5.86 


14.86 


5.93 


16 


17 


15.87 


6.09 


15.84 


6.16 


15.82 


6.23 


15.79 


6.30 


17 


18 


16.80 


6.45 


16.78 


6.52 


16.75 


6.60 


16.72 


6.67 


18 


19 


17.74 


6.81 


17.71 


6.89 


17.68 


6.96 


17.65 


7.04 


19 


20 
21 


18.67 


7.17 


18.64 


7.25 


18.61 


7.33 


18.58 


7.41 


20 
21 


19.61 


7.53 i 


19.57 


7.61 


19.54 


7.70 


19.50 


7.78 


22 


20.54 


7.88' 


20.50 


7.97 


20.47 


8.06 


20.43 


8.15 


22 


23 


21.47 


8.24i 


21.44 


8.34 


21.40 


8.43 


21.36 


8.52 


23 


24 


22.41 


8.60 


22.37 


8.70 


22.33 


8.80 


22.29 


8.89 


24 


25 


23.34 


8.96 


23.30 


9.06 


23.26 


9.16 


23.22 


9.26 


25 


26 


24.27 


9.32 


24.23 


9.42 


24.19 


9.53 


24.15 


9.63 


26 


27 


25.21 


9.68 


25.16 


9.79 


25.12 


9.90 


25.08 


10.01 


27 


28 


26.14 


10.03 


26.10 


10.15 


26.05 


10.26 


26.01 


10 38 


28 


29 


27.07 


10.39 


27.03 


10.51 


26.98 


10.63 


26.94 


10.75 


29 


30 


28.01 


10.75 


27.96 


10.87 


27.91 


11.00 


27.86 


11.12 


30 
31 


31 


28.94 


11.11 


28.89 


11.24 


28,84 


11.36 


28.79 


11.49 


32 


29.87 


11.47 


29.82 


11.60 


29.77 


11.73 


29.72 


11.86 


32 


33 


30.81 


11.83 


30.76 


11.96 


30.70 


12.09 


30.65 


12.23 


33 


34 


31.74 


12.18 


31.69 


12.32 


31.63 


12.46 


31.58 


12.60 


34 


35 


32.68 


12.54 


32.62 


12.69 


32.56 


12.83 


32.51 


12.97 


35 


36 


33.61 


12.90 


33.55 


13.05 


33.50 


13.19 


33.44 


13.34 


36 


37 


34.54 


13.26 


34.48 


13.41 


34.43 


13.56 


34.37 


13.71 


37 


38 


85.48 


13.62 


35.42 


13.77 


35.36 


13.93 


35.29 


14.08 


38 


39 


36.41 


13.98 


36.35 


14.14 


36.29 


14.29 


36.22 


14.45 


39 


40 

41 


37.34 


14.33 


37.28 


14.50 


37.22 


14.66 


37.15 


14.82 


40 
41 


38.28 


14.69 


38.21 


14.86 


38.15 


15.03 


38.08 


15.19 


42 


39.21 


15.05 


39.14 


16.22 


39.08 


15.39 


39.01 


15.56 


42 


43 


40.14 


15.41 


40.08 


15.58 


40.01 


15.76 


39.94 


16.93 


43 


44 


41.08 


15.77 


41.01 


15.95 


40.94 


16.13 


40.87 


16.30 


44 


45 


42.01 


16.13 


41.94 


16.31 


41.87 


16.49 


41.80 


16.68 


45 


46 


42.94 


16.48 


42.87 


16.67 


42.80 


16.86 


42.73 


17.05 


46 


47 


43.88 


16.84 


43.80 


17.03 


43.73 


17.23 


43.65 


17.42 


47 


48 


44.81 


17.20 


44.74 


17.40 


44.66 


17.59 


44.58 


17.79 


48 


49 


45.75 


17.56 


45.67 


17.76 


45.59 


17.96 


45.51 


18.16 


49 


50 

8 


46.68 


17.92 


46.60 


18.12 
Lat. 


46.62 


18.33 


46.44 


18.63 
Lat. 


50 

8 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 


Dep. 


1 



















Q 
















69] 


Deg. 


68| 


Deg 


681 


Deg. 


6Q\ 


Dog. 



T&AVERSE TABLE. 



115 



a 


21 Deg 


2U Deg. 


21| Deg. 


211 Deg. O 


1 










^ 














Hi I 


• 


Lat. 


Dep. 


Lat. 
47.53 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 3 
18 90 51 


51 


47.61 


18.28 


18.48 


47.45 


18.69 


47.37 


52 


48.55 


18.64 


48.46 


18 85 


48.38 


19.06 


48.30 


19.27 i 52 


53 


49.48 


18.99 


49.40 


19.21 


49.31 


19.42 


49.23 


19.64 


53 


54 


50.41 


19.35 


50.33 


19.57 


50.24 


19.79 


50.16 


20.01 


54 


55 


51 35 


19.71 


51.26 


19.93 


51.17 


20.16 


51.08 


20.38 


55 


56 


52 28 


20.07 


52.19 


20.30 


52.10 


20.52 


52.01 


20.75 


56 


57 


53 2] 


20.43 


53.12 


20.66 


53.03 


20.89 


52.94 


21.12 


5? 


58 


54.15 


20.79 


54.06 


21.02 


53.96 


21.26 


53.87 


21.49 


58 


59 


55.08 


21.14 


54.99 


21.38 


54.89 


21.62 


54.80 


21.86 


59 


60 


56.01 


21.50 


55.92 


21.75 


55.83 


21.99 


55.73 


22.22 


60 
61 


61 


56.95 


21.86 


56.85 


22.11 


56.76 


22.36 


56.66 


22.60 


62 


57, 88 


22.22 


57.78 


22.47 


57.69 


22.72 


57.59 


22.97 


62 


63 


58.82 


22.58 


58.72 


22.83 


58.62 


23.09 


58.52 


23.35 


63 


64 


59.75 


22.94 


59.65 


23.20 


59.55 


23.46 


59.44 


23.72 


64 


65 


60.68 


23.29 


60.58 


23.56 


60.48 


23.82 


60.37 


24.09 


65 


66 


61.62 


23.65 


61.51 


23.92 


61.41 


24.19 


61.30 


24.46 


66 


67 


62.55 


24.01 


62.44 


24.28 


62.34 


24.56 


62.23 


24.83 i 67 1 


68 


63.48 


24.37 


63.38 


24.65 


63.27 


24.92 


63.16 


25.20 


68 


69 


64.42 


24.73 


64.31 


25.01 


64.20 


25.29 


64.09 


25.57 


69 


70 


65.35 


25.09 


65.24 


25.37 


65.13 


25.66 


65.02 


25.94 


70 
71 


7) 


66.28 


25.44 


66.17 


25.73 


66.06 


26.02 


65.95 


26.31 


72 


67.22 


25.80 


67.10 


26.10 


66.99 


26.39 


66.87 


26.68 


72 


73 


68.16 


26.16 


68.04 


26.46 


67.92 


26.75 


67.80 


27.05 


73 


74 


69.08 


26.52 


68.97 


26.82 


68.85 


27.12 


68.73 


27.42 


74 


75 


70.02 


26.88 


69.9^ 


27.18 


69.78 


27.49 


69.66 


27.79 


75 


76 


70.95 


27.24 


70.83 


27.55 


70.71 


27.85 


70.59 


28.16 


76 


77 


71.89 


27.59 


71.76 


27.91 


71.64 


28.22 


71.52 


28.53 


77 


78 


72.82 


27.95 


72.70 


28.27 


72.57 


28.59 


72.45 


28.90 


78 


79 


73.75 


28.31 


73.63 


28.63 


73.50 


28.95 


73.38 


29.27 


79 


80 


74.69 
75.62 


28.67 


74.56 


29.00 


74.43 


29.32 


74.30 


29.64 


80 
81 


81 


29.03 


75.49 


29.36 


75.36 


29.69 


75.23 


30.02 


82 


76.55 


29.39 


76.42 


29.72 


76.29 


30.05 


76.16 


30.39 


82 


83 


77.49 


29.74 


77.36 


30.08 


77.22 


30.42 


77.09 


30.76 


83 


84 


78.42 


30.10 


78.29 


30.44 


78.16 


30.79 


78.02 


31.13 


84 


85 


79.35 


30.46 


79.22 


30.81 


79.09 


31.15 


78.95 


31.50 


85 


86 


80.29 


30.82 


80.15 


31.17 


80.02 


31.52 


79.88 


31.87 


86 


87 


81.22 


31.18 


81.08 


31.53 


80.95 


31.89 


80.81 


32,24 


87 


88 


82.16 


31 54 


82.02 


31.89 


81.88 


32.25 


81.74 


32.61 


88 


89 


83.09 


31.89 


82.95 


32.26 


82.81 


32.62 


82.66 


32.98 


89 


90 


84.02 


32.25 


83.88 


32.62 


83.74 
84.67 


32.99 


83.59 


33.35 
33.72 


90 

91 


91 


84.96 


32.61 


84.81 


32.98 


33.35 


84.52 


92 


85.89 


32.97 


85.74 


33.34 


85.60 


33.72 


85.45 


34.09 


92 


93 


86.82 


33.33 


86.68 


33.71 


86.53 


34.08 


186.38 


34.46 , 93 


94 


87.76 


33.69 


87.61 


34.07 


87.46 


34.45 


! 87.31 


34.83 1 94 


95 


88.69 


34.04 


88.54 


34.43 


88.39 


34.82 


i 88.24 


35.20 . 95 


96 


89.62 


34.40 


89.47 


34.79 


89.32 


35.18 


89.17 


35.57 


i 96 


97 


90.56 


34.76 


90.40 


35.16 


90.25 


35.55 


90.09 


35.94 


97 


98 


91.49 


35.12 


91.34 


35.52 


91.18 


35.92 


91.02 


36.31 


98 


99 


92.42 


35.48 


92.27 


35.88 ;1 92.11 


36.28 


91.95 


36,69 99 1 


lOO 

i 

§ 

♦J 

.2 
Q 


93.36 
Dop. 


35.84 


93.20 



Dep. 


36.24 93.04 


36.65 


92.88 


37.06 


100 


Lat. 


Lat. 


1 Dep. 


Lat. 


Dep. 


Lat. 


i d 
c 

c 

(5 


69 Deg. 


681 Deg. 


1 68^ Deg. 

1 


68i Deg 



22 



TRATERSE TABLE. 



C 22 Deg. 


22i Dcg. 


22^ 


Deg. 


221 Dog. 


"1 


9 














s 


S 
O 

9 


Lai. 


Dep. 


Lat. Dep. 


Lai. 
0.92 


Dep. '' 
0.33 


t Lai, 
0.92 


Dep. 
0.39 1 


1 


0.93 


0.37 


0.93 


0.38 


1 


2 1.35 


0.75 


1.85 


0.76 


1.85 


0.77 


1.84 


0.77! 


2 


3 2.78 


1.12 


2.73 


1.14 


2.77 


1.15 


2.77 


1.16 


3 


4 


3.71 


1.50 


3.70 


1.51 


3.70 


1.53 


3.69 


'.55 


4 





4.64 


1.37 


4.63 


1.89 


4.62 


1.91 


4.61 


i.93 


5 


6 


5.56 


2 . 25 


0. 00 


2.27 


5.. 54 


2.30 


5.53 


2 . 32 


6 


7 


6.49 


2 . 62 


6.48 


2.65 


6.47 


2.63 


6.46 2.71 1 


7 


fl 


7.42 


3.00 


7.40 


3.03 


7.39 


3.06 


7 38 3.09 


8 


9 


8.34 


3.37 


3 . 33 


3.41 


3.31 


3.44 


3.30 


3.48 


9 


10 
11 


9.27 


3.75 


9.26 


3.79 


9.24 


3.83 


9 . 22 


3.87 


iO 
" ll 


10.20 


4.12 


10.18 


4.17 


10.16 


4.21 


10.14 


4.25 


12 11.13 


4.50 


11.11 


4.54 


11.09 


4.59 


11.07 


4.64 


12 


13 


12.05 


4.37 


12.03 


4.92 


12.01 


4.97 


11.99 


5.03 


13 


14 


12.93 


5.24 


12.96 


5.30 


12.93 


5.36 


12.91 


5.41 


14 


15 


13.91 


5.62 13.33 


5.68 


13.86 


5.74 


13.33 


5.80 


15 


16 


14.33 


5.99 14.31 


6.06 


14.78 


6.12 


14.76 


6.19 


16 


17 


15.76 


6.37 15.73 


6.44 


15.71 


6.51 


15.63 


6.57 


17 


18 


16.69 


6.74 16.66 


6.32 


16.63 


6.39 


16.60 


6.96 


18 


19 


17.62 


7.12 


17.59 


7.19 


17.55 


7.27 


17.52 


7.35 


19 


20 

21 


18.54 


7.49 


13.51 


7.57 


13.43 


7.65 


13.44 


7.73 


20 


13.47 


7.87 


19.44 


7.95 


19.40 


8.04 


19.37 


8.12 


21 


22 


20.40 


8.24 


20 . 36 


8 . 33 


20 . 33 


8.42 


20.29 


8.51 


22 


23 


21.33 


3 . 62 


21.29 


3.71 


21.25 


8.80 


21.21 


3.39 


23 


24 


22 . 25 


8.99 


22 .21 


9.09 


22.17 


9.18 


22.13 


9.23 


24 


25 


23.18 


9.37 


23 . 14 


9.47 


23.10 


9.57 


23.05 


9.67 


25 


26 


24.11 


9.74 


24 06 


9.34 


24.02 


9.95 


23.93 


10.05 


26 


27 


25.03 


10.11 


24.99 


10.22 


24.94 


10.33 


24.90 


10.44 


27 


25 


25.96 


10.49 


25.92 


10.60 


25.37 


10.72 


25.32 


10.33 


23 


20 


26.39 


10.36 26.34 


10.93 


26.79 


11.10 


26.74 


11.21 


29 


30 27.82 

31 23.74 


11.24 


27.77 


11.36 


27.72 


11.43 


27.67 


11.60 


30 


11.61 


23.69 


11.74 


23 . 64 


11.36 


28.59 


11.99 


31 


32 29.67 


11.99 


29.62 


12.12 


29 . .56 


12.25 


29.51 


12.37 


32 


33 30.60 


12.36 


30 . 54 


12.50 


30 . 49 


12.63 


.30.43 


12.76 


33 


34 31.52 


12.74 


31.47 


12.37 


31.41 


13.01 


31.35 


13.15 


34 


35 32.45 


13.11 


32.39 13.25 


32.34 


13.39 


32.28 


13.53 


3C 


36 33.33 


13.49 


33.32 


13.63 


33.26 


13.73 


33.20 


13.92 


36 


37 34.31 


13.36 


34.24 


14.01 


34.18 


14.16 


34.12 


14.31 


37 


33 35 . 23 


14.24 


35.17 


14.39 


35.11 


14.. 54 


35.04 


14.70 


33 


39 36.16 


14.61 


36.10 


14.77 


36.03 


14.92 


35.97 


15.03 


39 


40 


37.09 


14. 9S 


37.02 


15.15 


36.96 


15.31 


36.39 

37.31 


15.47 
15. 36 


40 

' 41 


41 


33.01 


15.36 


37.95 


15.52 


37.33 


15- 69 


42 33.94 


15.73 


33.37 


15.90 


33.30 


16.07 


33 . 73 


16.24 


42 


43 39.37 


16.11 


39.30 


16.23 


39.73 


16.46 


39.65 


16.63 


43 


U 40.80 


16.43 


40.72 


16.66 


40.65 


16.84 


40 . 58 


17.02 


44 


45 41.72 


16.86 


41.65 


17.04 


41.57 


17.22 


41.50 


17.40 


45 


46 42.65 


17.23 


42.57 


17.42 


12 . 50 


17.60 


42.42 


17.79 


46 


47 43.53 


17.61 


43.50 


17.30 


43.42 


17.99 


43 . 3-1 


13.13 


47 


48 44.50 


17.93 


44.43 


13.18 


44.35 


18.37 


44.27 


18.56 


4^ 


49 45.43 


13.36 


45.35 


18.55 


45.27 


18.75 


45.19 18.95 


49 


50 46.36 
i j Dep. 


18.73 


46.23 


13.93 


46.19 


19.13 


46.11 
Dep. 


19.34 
Lat. 


50 

d 

c 
S 


Lat, 


, Dep. 


Lai. 


Dep. 
671 


Lat. 
Deg. 


3 

5 


68] 


67} Deg. 


67} Deg. 



TRAVERSE TABLE. 



117 



a 

S' 

f 

o 

? 


22Deg. 


22{ Dog. 


22| Deg. 


22| Deg. 


a 
1 

a 
tl 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


61 


47.29 


19.10 


47.20 


19.31 


47.12 


19.52 


47.03 


19.72 


62 


48.21 


19.48 


48.13 


19.69 


48.04 


19.90 


47.95 


20.11 


62 


63 49.141 


19.86 


49.05 


20.07 


48.97 


20.28 


48.88 


20.60 


63 


64 60.071 


20.23 


49.98 


20.45 


49.89 


20.66 


49.80 


20.88 


54 


65 


61.00 


20.60 


50.90 


20.83 


50.81 


21.05 


.50.72 


21.27 


55 


66 


61.92 


20.98 


51.83 


21.20 


51.74 


21.43 


51.64 


21.66 


66 


57 


52.86 


21.35 


52.76 


21.58 


52.66 


21.81 


52.57 


22.04 


57 


68 


63.78 


21.73 


63.68 


21.96 


53.59 


22.20 


53.49 


22.43 


58 


59 


54.70 


22.10 


54.61 


22.34 


54.51 


22.58 


54.41 


22.82 


59 


60 
61 


55.63 


22.48 


56.53 


22.72 


65.43 
56.36 


22.96 


55.33 


23.20 


60 
61 


56.56 


22.85 


56.47 


23.10 


23.34 


56.25 


23.69 


62 


57.49 


23.23 


57.38 


23.48 


57.28 


23.73 


57.18 


23.98 


62 


63 


58.41 


23.60 


58.31 


23.85 


58.20 


24.11 


58.10 


24.38 


63 


64 


59.34 


23.97 


.59.23 


24.23 


59.13 


24.49 


59.02 


24.75 


64 


65 


60.27 


24.35 


60.16 


24.61 


60.05 


24.87 


59.94 


25.14 


65 


66 


61.19 


24.72 


61.09 


24.99 


60.98 


25.26 


60.87 


25.52 


66 


67 


62.12 


25.10 


62.01 


25.37 


61.90 


25.64 


61.79 


25.91 


67 


68 


63.05 


25.47 


62.94 


25.75 


62.82 


26.02 


62.71 


26.30 


68 


69 


63.98 


25.85 


63.86 


26.13 


63.75 


26.41 


63.63 


26.68 


69 


70 
71 


64.90 


26.22 


64.79 


26.51 


64.67 


26.79 


64.55 


27.07 


70 
71 


65.83 


26.60 


65.71 


26.88 


65.60 


27.17 


65.48 


27.46 


72 


66.76 


26.97 


66.64 


27.26 


66 52 


27.55 


66.40 


27.84 


72 


73 


67.68 


27.35 


67.56 


27.64 


67.44 


27.94 


67.32 


28.23 


73 


74 


68.61 


27.72 


68.49 


28.02 


68.37 


28.32 


68.24 


28.62 


74 


75 


69.54 


28.10 


69.42 


28.40 


69.29 


28.70 


69.17 


29.00 


75 


76 


70.47 


28.47 


70.34 


28.78 


70.21 


29.08 


70.09 


29.39 


76 


77 


71.39 


28.84 


71.27 


29.16 


71.14 


29.47 


71.01 


29.78 


77 


78 


72.32 


29.22 


72.19 


29.53 


72.06 


29.85 


71.93 


30.16 


78 


79 


73.25 


29.59 


73.12 


29.91 


72.99 


30.23 


72.85 


30.55 


79 


80 
81 


74.17 


29.97 


74.04 


30.29 


73.91 


30.61 


73.78 


30.94 


80 
81 


75.10 


30.34 


74.97 


30.67 


74.83 


31.00 


74.70 


31.32 


82 


76.03 


30.72 


75.89 


31.05 


75.76 


31.38 


75.62 


31.71 


82 


83 


76.96 


31.09 


76.82 


31.43 


76.68 


31.76 


76.54 


,32.10 


83 


84 


77.88 


31.47 


77.75 


31.81 


77.61 


32.15 


77.46 


32.48 


84 


85 


78.81 


31.84 


78.67 


32.19 


78.53 


32.53 


78.39 


32.87 


85 


86 


79.74 


32.22 


79.60 


32.56 


79.45 


32.91 


79.31 


33.26 


86 


87 


80.60 


32.59 


80.52 


32.94 


80.38 


33.29 


80.23 


33.64 


87 


88 


81.69 


32.97 


81.45 


33.32 


81.30 


33.68 


81.15 


34.03 


88 


89 


82.52 


33.34 


82.37 


33.70 


82.23 


34.06 


82.08 


34.42 


89 


90 


83.45 


33.71 


83.30 


34.08 


83.16 


34.44 


83.00 


34.80 


90 
91 


91 


84.37 


34.09 


84.22 


34.46 


84.07 


34.82 


83.92 


35.19 


9i 85.30 


34.46 


85.16 


34.84 


85.00 


35.21 


84.84 


36.68 


92 


93 86.23 


34.84 


86.08 


35.21 


85.92 


35.59 


85.76 


35.96 


93 


94 87.16 


35.21 


87.00 


36.59 


86.84 


35.97 


86.69 


36.35 


94 


95 88. 08 


36.59 


87.93 35.97 


87.77 


36.35 


87.61 


36.74 


1 9& 


96 89.01 


36.96 


88.86 36 35 


88.69 


36.74 


88.53 


37.12 


1 96 


M- 89.94 


36.34 


89.78 


36.73 


89.62 1 37.12 


89.45 


37 51 


97 


9S 90 86 


36.71 


90.70 


37.11 


90.54 


37.50 


90.38 


37.90 


98 


9r, 91.79 


37.09 


91.63 


37.49 


91.46 


37.89 


91.30 


38.28 


99 


)00 


92 72 


37.46 


92.65 


37.86 


92.39 
Dep. 


38.27 


92.22 


38.67 
Lat. 


100 

i 


S 


Dep. 


Lat. 


Dep. 1 Lat. 


Lat. 


Dep. 




68 Deg. 


671 Deg. 


er^Deg. 


(JTi Deg. 



118 



TRAVEESE TABUE. 



? 


ij 
23 Deg. 1 23i 


Deg. 


23^ Deg. m Deg. 


w 


g 


ii 








? 


' ,-v i' 








? 


LaU ! Dep. Lat. 


Dep. 


; Lat. j 


Dep. Lat. Dep. • 


ft 
a 


1 


0.92 0.39 0.92 


0.39 


0.92 


0.40 0.92 0.40, 


2 1.84 0.?8 1.84 


0.79 1.83 


0.80 1.83 0.81 


2 


3 2.76 1.17 2.76 


1.13 2,75 


1.20 2.75 1.21 


3 


4 S.63 1.56 3.6S 


1.5-3 3.67 


1.59 3.66 1.61; 


i 


6 4.60 1.95 4.59 


1.97 4.59 


1.99 4.53 2.011 


T 


6 5.52 2.34 5.51 


2.37' 5.50 


2.39 5.49 2.42 1 


6 


7 6.-i4 2 74 6.43 


2.76 1 6.42 


2.79' 6.41 2.82 


7 


8 7.36 3.13 7.35 


3.16 7.34 


3.19 7.32 3.22 


8 


9 8.28 3.52 8.27 


3.55 8.25 


3.59 8.24 3.62 


9 


10 9.20 3.91 9.19 


3.95 9.17 


3.99 9.15 4.03 





11 10.13 4.30 10.11 


4. .34 10.09 


4.39 10.07 4.43 


11 


12 11.05 4.69 11.03 


4.74 11.00 


4.78 10.93 4.33' 


12 


13 11.97 5.03 11.94 


5.13 li.92 


5.18 11.90 6.24 


13 


14 12.89 5.47 12.86 


5.53 12.84 


5.58 12.81 5.64 


14 


15 13.81 5.86 13.78 


5.92 13.76 


5.98 13.73 6.04 


15 


16 14.73 6.25 14.70 


6.32 14.67 


6.38 14.64 6.44 


16 


17 15.65 6.64 15.62 


6.71 15.59 


6.78 15.56 6.S5 


17 


IS 16.57 7.03 16.54 


7.11 16.51 


7.13 16.48 7.25 


18 


19 )7.49 7.42 17.46 


7.50 17.42 


7.55 17.39 7.65 


19 


20 


IS. 41 • 7.81 18.38 
19.33 8.21 ' 19.29 


7.S9 18.34 
8.29 19.26 


7.97 18.31 8.05 
8.37 19.22 8.46 


20 

21 


21 


22 20.25 8.60 20.21 


S.6S 20.13 


8.77 20.14 S.S6 


22 


23 21.17 8.99 21.13 


9. OS 21.09 


9.17 21.05 9.26 


23 


24 22.09 9.38 ' 22.05 


9.47 22.01 


9.57 21.97 9.67 


24 


25 23.01 9.77 22.97 


9.87 22.93 


9.97 22.83 10.07 


25 


26 23.93 ^0.16 23.89 


10.26 23.84 


10.37 23.80 10.47 


26 


27 24.85 .0.55 24.81 


10.66 24.76 


10.77 24.71 10.87 


27 


23 25.77 .0.94 25.73 


11.05 25.68 


11.16 25.63 11.23 


.T£5 


29 26.69 U.33 26.64 


11.45 26.59 


11.56 26.54 11.63 


29 


30 


27.62 11.72 27.56 


11.84 


27.51 
2S.43 


11.96 27.46 12.08 
12.36 28.37 12.49 


30 


31 


23.54 12.11 2^.4S 


12.24 


31 


32 29.46 12.50 29. 4U 


12.63 29.35 


12.76 29.29 12.89 


a2 


33 30.38 12.89 30.32 


13.03 30.26 


13.16 30.21 13.29 


33 


34 31.30 13.28 31.24 


'13.42 31.13 


13.56 31.12 13.69 


34 


35 32.22 13.68 32.16 


13.82 32.10 


13.96 32.04 14.10 


35 


36 33.14 14.07 33.03 


14.21 33.01 


14.35 32.95 14.50 


36 


37 34.06 14.46 34.00 


14.61 33.93 


14.75 33.37 14.90 


37 


3S 34.93 14.85 34.91 


15.00 34.85 


15.15 34. 7S 15.30 


.38 


39 35.90 15.24 35.83 


15.39 35.77 


15.55 35.70 15.7] 


39 


40 36. S2 15.63 36.75 


15.79 36.68 


15.95 36.61 16.11 


40 


41 37.74 16.02 37.67 


16.13 37.60 


16.35 37.53 16.51 


11 


42 3S.66 16.41 3S.59 


16. 5S 3S.52 


16.75 3S.44 16.92 


42 


43 39. 5S 16. SO 39.51 


16.97 39.43 


17.15 39.36 17.32 


i i3 


41 40 50 17.19 40.43 


17.37 40.35 


17.54 40.27 17.72 


! 44 


4c 41 i2 17.58 41.35 


17.76 41.27 


17.94 4].:- l'.:2 


4.5 


46 42.34 17.97 42.26 


18.16 42.13 


18.34 42.:: l-..:3 


46 


47 4:3.26 18.36 43.18 


18.55 43.10 


15.74 43.02,15.93 


I 47 


48 41.18 18.76 44.10 


18.95 44.02 


19.14 43.93.19.3,3 


i 48 


t9 45.^0 19.15 4.5.02 


19. .34 44.94 


19.54 44.-5 19.73 


' 49 


50 4'..<;3 19.54 45.94 


19.74 45.85 


19.94 45.77,20.14 


, 50 


8 Dep. Lat. Dep. 
1 '" H 


Lat. 


Dep. 


Lat. Dop. Lat, 


5 


j; 




- 


67 Deg ' 66} 


Deg. 


66^ 


Deg. 66i Deg. 


it^ 



TRAVERSE TA.BLfc. 



119 



a 
a 
a 

51 


23 Deg. 


23i Deg. 


23^ Deg. 


23| Deg. 


a 

3 
o 
? 

51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 


46.95 


19.93 


46.86 


20.13 


46.77 


20.34 


46.68 


20.54 


52 


47 87 


20.32 


47.78 


20 . 53 


47.69 


20.73 


47.60 


20.94 


53 


53 


48,79 


20.71 


48.70 


20 . 92 


48.60 


21.13 


48.51 


21.35 


53 


54 


49.71 


21.10 


49.61 


21.. '52 


49.52 


21.53 


49 43 


21.75 


b4 


55 


50.63 


21.49 


50.53 


21.71 


50.44 


21.93 


50.34 


22.15 


55 


56 


51.55 


21.88 


51.45 


22.11 


51.36 


22.33 


51.26 22.55 


56 


57 


52.47 


22.27 


52.37 


22.50 


52.27 


22.73 


52.17 


22.96 57 


58 


53.39 


22 . 66 


53.29 


22.90 


53.19 


23.13 


53.09 


23.36 58 


59 


54.31 


23.05 


.54.21 


23 . 29 


54.11 


23.53 


54 00 


23.76 59 


60 

61 


55.23 


23.44 


55.13 


23.68 


55.02 


23.92 


54.92 


24.16 


60 


56.15 


23.83 


56.05 


24.08 


55.94 


24.32 


55.83 


24.57 


61 


62 


57.07 


24.23 


56.97 


24 47 


56.86 


24.72 


56.75 


24.97 


62 


63 


57.99 


24.62 


57.88 


24.87 


57.77 


25. J 2 


57.66 


25.37 


63 


64 


58.91 


25.01 


58.80 


25.26 


58.69 


25 . 52 


58.58 


25.78 


64 


65 


59.83 


25.40 


59.72 


25.66 


59.61 


25.92 


59.50 


26.18 


65 


66 


60.75 


25 . 79 


60.64 


26.05 


60.53 


26.32 


60.41 


26.58 


66 


67 


61.67 


26.18 


61.56 


26.45 


61.44 


26.72 


61.33 


26.98 


67 


68 


62.59 


26.57 


62.48 


26.84 


62.36 


27.11 


62.24 


27.39 


68 


69 


63.51 


26.96 


63.40 


27.24 


63.28 


27.51 


63.16 


27.79 


69 


70 
71 


64.44 


27.35 


64.32 
65.23 


27.63 


64.19 


27.91 


64.07 


28.19 


70 
71 


65.36 


27.74 


28.03 


65.11 


28.34 


64.99 


28.59 


72 


66.28 


28.13 


66.15 


28.42 


66.03 


28.71 


65.90 


29.00 


72 


73 


67.20 


28.52 


67.07 


28.82 


66 . 95 


29.11 


66.82 


29.40 


73 


74 


68.12 


28.91 


67.99 


29.21 


67.86 


29.51 


67.73 


29.80 


74 


75 


69.04 


29.30 


68.91 


29.61 


68.78 


29.91 


68.65 


30.21 


75 


76 


69.96 


29.ro 


69.83 


30.00 


69.70 


30.30 


69.56 


30.61 


76 


77 


70.88 


30.09 


70 . 75 


30.40 


70.61 


30.70 


70.48 


31.01 


77 


78 


71.80 


30.48 


n.67 


30.79 


71.53 


31.10 


71.39 


31.41 


78 


79 


72.72 


30.87 


72 . 58 


31.18 


72.45 


31.50 


72.31 


31.82 


79 


80 
81 


73.64 


31.26 


73.50 


31.58 


73.36 


31.90 


73.22 


32.22 


80 


74.56 


31.65 


74.42 


31.97 


74.28 


31i.30 


74.14 


32.62 


81 


82 


75.48 


32.04 


75.34 


32.37 


75.20 


32 . 70 


75.06 


33.03 


82 


83 


76.40 


32.43 


76.26 


32.76 


76.12 


33.10 


75.97 


33.43 


83 


84 


77.32 


32.82 


77.18 


33.16 


77.03 


33.49 


76.89 


33.83 


84 


85 


78.24 


33.21 


78.10 


33.55 


77.95 


.33.89 


77.80 


34.23 


85 


86 


79.16 


33 . 60 


79.02 


33.95 


78.87 


34.29 


78.72 


34.64 


86 


87 


80.08 


33.99 


79.93 


34.34 


79.78 


34.69 


79.63 


35.04 


87 


88 


81.00 


34.38 


80.85 


34.74 


80.70 


35.09 


80.55 


35.44 


88 


89 


81.92 


34.78 


81.77 


35.13 


81.62 


35.49 


81.46 


35.84 


89 


90 
91 


82.85 
83.77 


35-17 


82.69 


35.53 


82.54 


35.89 


82.38 


36.25 


90 
'91 


35.56 


83.61 


35.92 


83.45 


36.29 


83.29 


36.65 1 


92 


84.69 


35.95 


84.53 


36.32 


84.37! 36.68 


84.21 


37.05 92 


93 


85.61 


36.34 


85.45 


36.71 


85.29 37.08 


85.12 


37.46 , 93 


94 


86.53 


36.73 


86.37 


37.11 


86.20 37.48 


86.04 


37.86 ' 94 


95 


87.45 37.12 


87.29 


37.50 


87.12 37.88 


86.95 


38.26 95 


96 88.37 37.51 


88.20 


37.90 


88.04 38.28 


87.87 


38.66 ; ^6 


97 89.29 37.90 


89.12 


38.29 


88.95 38.68 


88.79 


39.07 i )1 


98 90.21 


38.29 


90.04 


33.68 


89.87 39.08 


89.70 


39.47 98 


99 91.13 


38.68 


90.96 


39.08 


90.79 


39.48 


90.62 


39.87 


99 


100 92.05 


39.07 


91.88 


39.47 


91.71 


39.87 


91.53 


40.27 


100 


i\ 


Dep. 


Lat. 


Dep. 


Lat. 


Dep 


Lat. 


Dep 


Lat. 


i 


s 


67 Deg. 


66| Deg. 


661 Deg. 


... .| 
m Deg. : 


,1 






, . „> 



120 



TRAVERSE TABLE. 



CD 


24 Dog. 


24i Deg. 


24A Dog. 


1 

S4| Deg. 


1 


g 

o 
o 










1 

1 


Lat. 1 Dep. 


Lat. 


Dep. 


Lat, 


Dop. 


Lat. 


Dep. 


I 


0.91 1 0.41 


0.91 


0.41 


0.91 


0.41 


0,91 


0.42 


2 


1.83 0.81 


1.82 


0.82 


1.82 


0.83 


1.82 


0.84 


2 


3 


2.74 1.22 


2.74 


1.23 


2.73 


1.24 


2 72 


1.26 


3 


4 


3.65 1.63 


3.65 


1.64 


3.64 


1.66 


3.63 


1.67 


4 


6 


4.67 


2.03 


4.56 


2.05 


4.55 


2.07 


4.54 


2.09 


5 


6 


5.48 


2.44 


5.47 


2.46 


5.46 


2.49 


6.45 


2.51 


6 


7 


6.39 


2.85 


6.38 


2.87 


6.37 


2.90 


6.36 


2.93 


7 


8 


7.31 


3.25 


7.29 


3.29 


7.28 


3.32 


7.27 


3.35 


8 


9 


8.22 


3.66 


8.21 


3.70 


8.19 


3.73 


8.17 


3.77 


9 


10 
11 


9.14 


4.07 
4.47 


9.12 


4.11 


9.10 


4.15 


9.08 


4.19 


/O 

11 


10.05 


10.03 


4.52 


10.01 


4.56 


9.99 


4.61 


12 


10.96 


4.88 


10.94 


4.93 


10.92 


4.98 


10.90 


5.02 


12 


13 


11.88 


6.29 


11.85 


5.34 


11.83 


5.39 


11.81 


5.44 


13 


14 


12.79 


5.69 


12.76 


5.75 


12.74 


5.81 


12.71 


5.86 


14 


15 


13.70 


6.10 


13.68 


6.16 


13.65 


6.22 


13.62 


6.28 


15 


16 


14.62 


6.51 


14.59 


6.57 


14.56 


6.64 


14.53 


6.70 


16 


17 


15.53 


6.92 


15.50 


6.98 


15.47 


7.05 


15.44 


7.12 


17 


18 


16.44 


7.32 


16.41 


7.39 


16.38 


7.46 


16.35 


7.54 


18 


19 


17.36 


7.73 


17.32 


7.80 


17.29 


7.88 


17.25 


7.95 


19 


20 
21 


18.27 


8.13 


18.24 


8.21 


18.20 


8.29 


18.16 


8.37 


20 

21 


19.18 


8.54 


19.15 


8.63 


19.11 


8.71 


19.07 


8.79 


22 


20.10 


8.95 


20.06 


9.04 


20.02 


9.12 


19.98 


9.21 


22 


23 


21.01 


9.35 


20.97 


9.45 


20 . 93 


9.54 


20.89 


9.63 


23 


24 


21.93 


9.76 


21.88 


9.86 


21.84 


9.95 


21.80 


10.05 


24 


25 


22.84 


10.17 


22.79 


10.27 


22.75 


10.37 


22.70 


10.47 


25 


26 


23.75 


10.58 


23.71 


10.68 


23.66 


10.78 


23.61 


10.89 


26 


27 


24.67 


10.98 


24.62 


11.09 


24.57 


11.20 


24.52 


11.30 


27 


28 


25.58 


11.39 


25.53 


11.50 


25.48 


11.61 


25.43 


11.72 


28 


29 


26.49 


11.80 


26.44 


11.91 


26.39 


12.03 


26.34 


12.14 


29 


30 
31 


27.41 


12.20 


27.35 


12.32 


27.30 


12.44 
12.86 


27.24 


12.. 56 


30 
31 


28.32 


12.61 


28.26 


12.73 


28.21 


28.15 


12.98 


32 


29.23 


13.02 


29.18 


13.14 


29.12 


13.27 


29.06 


13.40 


32 


33 


30.15 


13.42 


30.09 


13.55 


30.03 


13.68 


29.97 


13.82 


33 


34 


31.06 


13.83 


31.00 


13.96 


30.94 


14.10 


30.88 


14.23 


34 


35 


31.97 


14.24 


31.91 


14.38 


31.85 


14.51 


31.78 


14.65 


35 


36 


32.89 


14.64 


32.82 


14.79 


32.76 


14.93 


32.69 


15.07 


36 


37 


33.80 


15.05 


33.74 


15.20 


33.67 


15.34 


33.60 


15.49 


37 


38 


34.71 


15.46 


34.65 


15.61 


34.58 


15.76 


34.51 


15.91 


38 


39 


35.63 


15.86 


36.56 


16.02 


35.49 


16.17 


35.42 


16.33 


39 


40 
41 


36.54 


16.27 


86.47 


16.43 


36.40 
37.31 


16.59 
17.00 


36.33 16.75 
37.23 ; 17.16 


40 
41 


37.46 


16.68 


37.38 


16.84 


42 


38.37 


17.08 


38.29 


17.25 


38.22 


17.42 


38.14 


17.58 


12 


43 


39.28 


17.49 


39.21 


17.60 


39.13 


17.83 


39.05 


18.00 


id 


44 


40.20 


17.90 


40.12 


18.07 


40.04 


18.25 


39.96 


18.42 


44 


45 


41.11 


18 30 


41.03 


18.48 


40.95 


18.66 


40.87 


18.84 


45 


46 


42.02 


18.71 


41.94 


18.89 


41.86 


19.08 


41.77 


19.26 


46 


47 


42.94 


19.12 


42.85 


19.30 


42.77 


19.49 


42.68 


19,68 


47 


i8 


43.85 


19.52 


43.76 


19.71 


43.68 


19.91 


43.59 


20.10 


48 


49 


44.76 19.93 


44.68 


20.13 


44.59 


20.32 


44.50 


20.51 


19 


60 


45.68 120.34 


46.59 


20.54 


45.50 
Dep, 


20.73 


45.41 
Dep. 


20^93 
Lat. 


50 

V 

a 

d 

♦J 

CD 

5 


s 


Dop. 


Lat. 


Dep. 


Lat. 


Lat. 


66 Deg. 


65! Deg. 

1 


65^ 


D'^Sr. 


654 Deg 



TRAVEliSE TABLE. 



121 



P 

9 
o 
p 

61 


24Deg. 


24i Deg. 


24i Deg. 


241 De^,. 


9. 

n 
? 
51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


46.59 


20.74 


46.50 


20.95 


46.41 


21.15 


46.32 


21.35 


52 


47.50 


21.15 


47.41 


21.36 


47.32 


21.56 


47.22 


21.77 


52 


53 


48.42 


21.56 


48.32 


21.77 


48.23 


21.98 


48.13 


22.19 


53 


64 


49.33 


21.96 


49.24 


22.18 


49.14 


22.39 


49.04 


22.61 


54 


56 


50.24 


22.37 


50.15 


22.59 


50.05 


22.81 


49.95 


23.03 


56 


56 


51.16 


22.78 


51.06 


23.00 


50.96 


23.22 


50.86 


23.44 


66 


57 


52.07 


23.18 


51.97 


23.41 


51.87 


23.64 


51.76 


23.86 


57 


58 


52.99 


23.59 


52.88 


23.82 


52.78 


24.05 


52.67 


24.28 


58 


59 


53.90 


24.00 


53.79 


24.23 


53.69 


24.47 


53.58 


24.70 


59 


60 
61 


54.81 


24.40 


54.71 


24.64 


54.60 


24.88 


54.49 


26.12 


60 

61 


55.73 


24.81 


55.62 


25.05 


55.51 


25.30 


55.40 


25.54 


62 


56.64 


25.22 


56.53 


25.46 


56.42 


25.71 


56.30 


25.96 


62 


63 


57.55 


25.62 


57.44 


25.88 


57.33 


26.13 


57.21 


26.38 


63 


64 


.58.47 


26.03 


58.35 


26.29 


58.24 


26.54 


58.12 


26.79 


64 


65 


59.38 


26.44 


59.26 


26.70 


59.15 


26.96 


59.03 


27.21 


65 


66 


60.29 


26.84 


60.18 


27.11 


60.06 


27.37 


59.94 


27.63 


66 


67 


61.21 


27.25 


61.09 


27.52 


60.97 


27.78 


60.85 


28.05 


67 


68 


62.12 


27.66 


62.00 


27.93 


61.88 


28.20 


61.75 


28.47 


68 


69 


63.03 


28.06 


62.91 


28.34 


62.79 


28.61 


62.66 


28.89 


69 


70 

71 


63.95 


28.47 


63.8-2 


28.75 


63.70 


29.03 


63.57 


29.31 


70 
71 


64.86 


28.88 


64.74 


29.16 


64.61 


29.44 


64.48 


29.72 


72 


65.78 


29.28 


65.65 


29.57 


65.52 


29.86 


65.39 


30.14 


72 


73 


66.69 


29.69 


66.56 


29.98 


66.43 


30.27 


66.29 


30.. 56 


73 


74 


67.60 


30.10 


67.47 


30.39 


67.34 


30.69 


67.20 


30.98 


74 


75 


68.52 


30.51 


68.38 


30.80 


68.25 


31.10 


68.11 


31.40 


75 


76 


69.43 


30.91 


69.29 


31.21 


69.16 


31.52 


69.02 


31.82 


76 


77 


70.34 


31.32 


70.21 


31.63 


70.07 


31.93 


69.93 


32.24 


77 


78 


71.26 


31.73 


71.12 


32.04 


70.98 


32.35 


70.84 


32.66 


78 


79 


72.17 


32.13 


72.03 


32.45 


71.89 


32.76 


71.74 


33.07 


79 


80 
81 


73.08 
74.00 


.32.54 


72.94 


32.86 


72.80 


33.18 


72.65 


33.49 


80 
81 


32.95 


73.85 


33.27 


73.71 


33.59 


73.56 


33.91 


83 


74.91 


33.35 


74.76 


33.68 


74.62 


34.00 


74.47 


34.33 


82 


83 


75.82 


33.76 


75.68 


34.09 


75.. 53 


34.42 


75.38 


34.75 


83 


84 


76.74 


34.17 


76.59 


34.50 


76.44 


34.83 


76.28 


35.17 


84 


85 


77.65 


34.57 


77.50 


34.91 


77.35 


35.25 


77.19 


35.59 


85 


86 


78.56 


34.98 


78.41 


35.32 


78 26 


35.66 


78.10 


36.00 


86 


87 


79.48 


35.39 


79.32 


35.73 


79.17 


36.08 


79.01 


36.42 


87 


88 


80.39 


35.79 


80.24 


36.14 


80.08 


36.49 


79.92 


36.84 


88 


89 


81.31 36.;sO 


81.15 


36.55 


80,99 


36.91 


80.82 


37 26 


99 


90 

9: 


82.22 36.61 


82.06 


36.96 


81.90 
82.81 


37.32 


81.73 


37.68 


90 
91 


83.13 37.01 


82.97 


37.38 


37.74 


82.64 


38.10 


92 


84.05 


37.42 


83.88 


37.79 


83.72 


38.15 


83.55 


38.. 52 


92 


93 84.96 


37.83 


84.79 


38.20 


84.63 


38.57 


84.46 


38.94 


93 


94 85 87 


38.23 


85.71 


38.61 


85.54 


38.98 


85.37 


39.35 


94 


95 86 79 


38.64 


86.62 


39.02 


86.45 


39.40 


86.27 


39 77 


95 


96 87.70 


39.05 1 


87.53 


39.43 


87.36 


39.81 


87.18 


40.19 


96 


97 88.61 


39. 45 1 


88.44 


39.84 


88.27 


40.23 


88.09 


40.61 


97 


98 Q9.53 139.06 1 


89.35 


40.26 


89.18 


40.64 


89.00 


41.03 


98 


99 


90.44 


40.27 


90.26 


40.66 


90.09 


41.05 


89.91 


41.45 


99 


tOO 


91.35 
Dep. 


40.67 
Lftt. 


91.18 
Dep. 


41.07 


91.00 


41.47 


90.81 


41.87 


100 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


66 Dt/T. 


65| Deg. 


65i Deg. 


65i Dog. 


\ 


:! ( 







122 



TRAVERSE TAJRLE. 



■ ' 
p 

3 

a 

a 

T 


25 Deg. 


25\ Deg. 


25i Deg 


251 Deg. 


§ 


Lat. 


Dop. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.91 


0.42 


9.90 


0.43 


0.90 


0.43 


0.90 


0.43 


2 


1.81 


0.85 


1.81 


0.85 


1.81 


0.86 


1.80 


0.87 


2 


3 


2.72 


1.27 


2.71 


1.28 


2.71 


1.29 


2.70 


1.30 


3 


4 


3.63 


1.69 


3.62 


1.71 


3.61 


1.72 


3.60 


1.74 


4 


6 


4.53 


2.11 


4.52 


3.13 


4.51 


2.15 


4.50 


2.17 


(> 


6 


5.44 


2.54 


5.43 


2.56 


5.42 


2.58 


5.40 


2.61 


6 


7 


6.34 1 2.96 


6., S3 


2.99 


6.32 


3.01 


6.30 


3.04 


7 


8 


7.25 3.38 


7.24 


3.41 


7.22 


3.44 


7.21 


3.48 


8 


9 


8.16 


3.80 


8.14 


3.84 


8.12 


3.87 


8.11 


3.91 


9 


10 
11 


9.06 
9.97 


4.23 


9.04 


4.27 


9.03 


4.31 


9.01 
" 9.91 


4.34 

4.78 


10 

11 


4.65 


9.95 


4.69 


9.93 


4.74 


12 


iO.88 


5.07 


10.85 


5.12 


10.83 


5.17 


10.81 


5.21 


12 


13 . 11.78 


5.49 


11.76 


5.55 


11.73 


5.60 


11.71 


5.65 


13 


14 


12.69 


5.92 


12.66 


5.97 


12.64 


6.03 


12.61 


6.08 


14 


16 


13.59 


6.34 


13.57 


6.40 


13.54 


6.46 


13.51 


6.52 


In 


16 


14.50 


0.76 


14.47 


6.83 


14.44 


6.89 


14.41 


6.95 


16 


17 


15.41 


7.18f 15.38 


7.25 


15.34 


7.32 


15.31 


7.39 


17 


18 


16.31 


7.61 16.28 


7.68 


16.25 


7.75 


16.21 


7.82 


18 


19 


17.22 


8.03! 17.18 


8.10 


17.15 


8.18 


17.11 


8.25 


19 


20 

21 


18.13 
19.03 


8.45 Ij 18.09 


8.53 


18.05 


8.61 


18.01 


8.69 
9.12 


20 
21 


8.871; 18.99 


8.96 


18.95 


9.04 


18.91 


22 


19.94 


9.30 19.90 


9.38 


19.86 


9.47 


19.82 


9.56 ! 


22 


23 


20.85 


9.72 20.80 


9.81 


20.76 


9.90 


20.72 


9.99 


23 


24 


21.75 


10.14 


21.71 


10.24 


21.66 


10.33 


21.62 


10.43 


24 


25 


22.66 


10.57 


22.61 


10.66 


22.56 


10.76 


22.52 


10.86 


25 


20 


23.56 


10.99 


23.52 


11.09 


23.47 


11.19 


23.42 


11.30 


26 


27 


24.47 


11.41 


24.42 


11.52 


24.37 


11.62 


24.32 


11.73 


27 


28 


25.38 


11.83 


25.32 


11.94 


25.27 


12.05 


25.22 


12.16 


28 


29 


26.28 


12.26 


26.23 


12.37 


26.17 


12.49 


26.12 


12.60 


29 


30 
31 


27.19 


12.68 


27.13 


12.80 


27.08 


12.92 


27.02 


13.03 


30 
31 


28.10 


13.10 


28.04 


13.22 


27.98 


13.35 


27.92 


13.47 


32 


29.00 


13.52 


28.94 


13.65 


28.88 


13.78 


28.82 


13.90 


32 


33 


29.91 


13.95 


29.85 


14.08 


29.79 


14.21 


29.72 


14.34 


33 


34 


30.81 


14.37 


30.75 


14.50 


30.69 


14.64 


30.62 


14.77 


34 


35 


31.72 


14.79 


31.66 


14.93 


31.59 


15.07 


31.52 


15.21 


35 


30 


32.63 


15.21 


32.56 


15.36 


32.49 


15.50 


32.43 


15.64 


36 


37 


33.53 


15.64 


33.46 


15.78 


33.40 


15.93 


33.33 


16.07 


37 


38 


34.44 


16.06 


34.37 


16.21 


34.30 


16.36 


34.23 


16.51 


3ft 


39 1 35.35 


16.48 


35.27 


16.64 


35.20 


16.79 


35.13 


16 94 


39 


40 
4- 


36.25 


16.90 


36.18 


17.06 


36.10 


17.22 
17.65 


36.03 


17.38 


40 
41 


37.16 


17.33 


37.08 


17.49 


37.01 


36.93 


17.81 


42 


38.06 


17.75 


37.99 


17.92 


37.91 


18.08 


37.83 


18.25 


42 


43 


38.97 


18.17 


38.89 


18.34 


38.81 


18.51 


38.73 


18.68 


43 


44 


39.88 


18.60 


39.80 


18.77 


39.71 


18.94 


39.63 


19.12 


44 


45 


40.78 


19.02 


40.70 


19.20 


40.62 


19.37 


40.53 


19. £5 


45 


46 


41.69 


19.44 


41.60 


19.62 


41.52 


19.80 


41.43 


19.98 


46 


47 


42,60 


19.86 


42.51 


20.05 


42.42 


20.23 


42.33 


20.42 


47 


4ft 


43.50 


20.29 


43.41 


20.48 


43.32 


20.66 


43.23 


20.86 


4fi 


49 


44.41 


20.71 


44.32 


20.90 


44.23 


21.10 


44.13 


21.29 ' 49 1 


50 45.32 


21.13 


45.22 


21.33 


45.13 


21.53 


45.03 


21.72 


60 


§ 

c 

d 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 


Lat. 


Dep. 


Let. 


d 

1 

.a 


65 Deg. 


64| 


Dea. 


64i Dog. 


64i Deg. 



TRAVERSE TARLR. 



123 



S' 

«— • 

p 

a 
a 
a 

61 


25 Dog. 


25i Dog 


25i Deg. 


251 Dog. 


1 

o 

1' 

? 

51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


46.22 


21.65 


46.13 


21.75 


46.03 


21.96 


46.94 


22.16 


52 


47.13 


21.98 


47.03 


22.18 


46.93 


22.39 


46.84 


22.59 


52 


53 


48.03 


22.40 


47.94 


22.61 


47.84 


22.82 


47.74 


23.03 


53 


54 


48.94 


22.82 


48.84 


23.03 


48.74 


23.25 


48.64 


23.46 


54 


55 


49.85 


23.24 


49.74 


23.46 


49.64 


23.68 


49.64 


23.89 


56 


66 


50.75 


23.67 


60.65 


23.89 


50.54 


24.11 


50.44 


24.33 561 


57 


51 66 


24.09 


61.65 


24.31 


51.45 


24.54 


51.34 


24.76 


57 


58 52.67 


24.51 


52.46 


24.74 


52.35 


24.97 


52.24 


25.20 


58 


69 


53.47 


24.93 


53.36 


25.17 


53.25 


25.40 


53.14 


25.63 


69 


60 
61 


54.38 


25.. 36 


54.27 


25.59 
26.02 


54.16 


25.83 


54.04 


26.07 


60 
~61 


55.28 


25.78 


55.17 


55.06 


26.26 


.54.94 


26.60 


62 


56.19 


26.20 


56.08 


26.45 


55.96 


36.69 


56.84 


2F.94 


62 


63 


57.10 


26.62 


56.98 


26.87 


56.86 


27.12 


.06 . 74 


5,7 37, 63 1 


64 


58.00 


27.05 


57.89 


27.30 


57.77 


27.56 


57.64 


27.80 


64 


65 


58.91 


27.47 


58.79 


27.73 


58.67 


27.98 


68.. 55 


28.24 


66 


66 


59.82 


27.89 


59.69 


28.15 


59.57 


28.41 


59.45 


28.67 


66 


67 


60.72 


28.32 


60.60 


28.58 


60.47 


28.84 


60.35 


29.11 


67 


68 


61.63 


28.74 


61.50 


29.01 


61.38 


29.27 


61.25 


29.64 


68 


69 


62.54 


29.16 


62.41 


29.43 


62.28 


29.71 


62.15 


29.98 


69 


70 

71 


63.44 


29.68 


63.31 


29.86 


63.18 


30.14 


63.06 


30.41 


70 
71 


64.35 


30,01 


64.22 


30.29 


64.08 


30.67 


63.95 


30.85 


72 


65.25 


.30.43 


65.12 


30.71 


64.99 


31.00 


64.85 


31.28 


72 


73 


66.16 


30.85 


66.03 


31.14 


65.89 


31.43 


65.75 


31.71 


73 


74 


67.07 


31.27 


66.93 


31.57 


66.79 


31.86 


66.65 


32.15 


74 


75 


67.97 


31.70 


67.83 


31.99 


67.69 


32.29 


67.55 


32.58 


75 


76 


68.88 


32.12 


68.74 


32.42 


68.60 


32.72 


68.45 


33.02 


76 


77 


69.79 


32.54 


69.64 


32.85 


69.50 


33.15 


69.35 


33.45 


77 


78 


70.69 


32.96 


70.55 


33.27 


70.40 


33.58 


70.25 


33.89 


78 


79 


71.60 


33.39 


71.45 


33.70 


71.30 


34.01 


71.16 


34.32 


79 


80 

81 


72.50 


33.81 


72.36 
73.26 


34.13 
34.55 


72.21 


34.44 


72.06 


34.76 
35.19 


80 
81 


73.41 


34.23 


73.11 


34.8/ 


72.96 


82 


74.32 


34.65 


74.17 


34.98 


74.0] 


35.30 


73.86 


35.62 


82 


83 


75.22 


35.08 


75.07 


35.41 


74. 9i 


35.73 


74.76 


36.06 


83 


84 


76.13 


35.50 


75.97 


35.83 


75.82 


36.16 


75.66 


36.49 


84 


85 


77.04 


35.92 


76.88 


36.26 


76.72 


36.59 


76.56 


36.93 


85 


86 


77.94 


36.35 


77.78 


36.68 


77.62 


,37.02 


77.46 


37.36 


86 


87 


78.85 


36.77 


78.69 


37.11 


78.52 


37.45 


78.36 


37.80 


87 


88 


79.76 


37.19 


79.59 


37.54 


79.43 


37.88 


79.26 


38.23 


88 


89 


80.66 


37.61 


80.50 


37.96 


80.33 


38.32 


80.16 


38.67 


89 


90 
91 


81.67 


38.04 


81.40 


38.39 


81.23 


38.75 


81.06 


39.10 


90 
91 


82.47 


38.46 


82.31 


38.82 


82.14 


39.18 


81.96 


39.63 


92 


83.38 


38.88 


83.21 


39.24 


83.04 


39.61 


82.86 


39.97 


92 


93 


84.29 


39.30 


84.11 


39.67 


83.94 


40.04 


83.76 


40.40 


93 


94 


85.19 


39.73 


85.02 


40.10 


84. S4 


40.47 


84.67 


40.84 


94 


96 


86.10 


40.15 


85.92 


40.52 


85.75 40.90 


85 67 41.27 


95 


96 


87.01 


40.57 


86.83 


40.95 


86.65 41.33 


86,47 


41.71 


96 


97 


87.91 


40.99 


87.73 


41.38 


87.55 


41.76 


87.37 


42.14 


97 


98 


88.82 


41.42 


88.64 


41.80 


88.45 


42.19 


88.27 


42.. 58 


98 


99 


89.72 


41.84 


89.64 


42.23 


89.36 


42.62 


89.17 


43.01 


99 


100 

o 

c 

5 


dO.63 
Dep. 


42.26 


90.46 


42.66 


90.26 


43.05 


90.07 


43 44 


100 

o 

e 
a 

.2 

O 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


65Deg. 


64f Deg. 


64i Deg. 


64i Deg. 



124 



TRAVElUsf lABIilC 



1 
S 1 

g 


26 Deg. 


26i Deg. 

t 


26i Deg. 

1 


261 Deg. 

1 


2 
s* 

9 

3 

? 

~l 


Lat. 


Dep. 
0.44 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.90 


0.90 


0.44 


0.89 


0.45 


0.80 


0„45l 


2 


1.80 


0.88 


1.79 


0.88 


1.79 


0.89 


1.79 


0.90 


2 


3 


3 70 


1.32 


2 69 


1.33 


2.68 


1.34 


2.68 


1.35 


3 


4 


3.60 


1 75 


3.59 


1.77 


3.58 


1.78 


3.57 


1.80 \ I 


E 


4.49 


2 19 


4.48 


2.21 


4.47 


2.23 


4.48 


2.25 


5 1 


6 


5.39 


2.63 


5.38 


2.85 


5.37 


2.68 


5.36 


2.70 


6 1 


7 


6.29 


3.07 


6.28 


3.10 


6.26 


3.12 


6.25 


3.15 


7 1 


8 


7.19 


3.51 


7.17 


3.54 


7.16 


3.57 


7.14 


3.80 


8 1 


9 


8.09 


3.95 


8.07 


3.98 


8.05 


4.02 


8.04 


4.05, 9 1 


10 

" 11 


8.99 


4.38 


8.97 


4.42 


8.95 


4.46 


8.93 


4.50 10 1 


9.89 


4.82 


9.87 


4.87 


9.84 


4.91 


9.82 


4.95 


11 


12 


10.79 


5.26 


10.76 


5.31 


10.74 


5.35 


10.72 


5.40 


12 


13 


11.68 


5.70 


11.66 


5.75 


11.63 


5.80 


11.61 


5.85 


13 


14 


12.58 


6.14 


12.58 


8.19 


12.53 


8.25 


12.50 


6.30 


14 


15 


13.48 


6.58 


13.45 


C 63 


13.42 


6.69 


13.39 


6.75 


15 


16 


14.38 


7.01 


14.35 


7.08 


14.32 


7.14 


14.29 


7.20 


16 


17 


15.28 


7.45 


15.25 


7.52 


15.21 


7.59 


15.18 


7.65 


17 


18 


16.18 


7.89 


16.14 


7.96 


18.11 


8.03 


18.07 


8.10 


18 


19 


17.08 


8.33 


17.04 


8.40 


17.00 


8.48 


16.97 


8.55 


19 


20 
21 


17.98 


8.77 


17.94 


8.85 
9.29 


17.90 


8.92 


17.86 


9.00 


20 

21 


18.87 


9.21 


18.83 


18.79 


9.37 


18.75 


9.45 


22 


19.77 


9.64 


19.73 


9.73 


19.09 


9.82 


19.65 


9.90 


22 


23 


20.67 


10.08 


20.63 


10.17 


20.58 


10.28 


20.. 54 


10.35 


23 


24 


21.57 


10.52 


21.52 


10.61 


21.48 


10.71 


21.43 


10.80 


24 


25 


22.47 


10.96 


22.42 


11.06 


22.37 


11.15 


22.32 


11.25 


25 


26 


23.37 


11.40 


23.32 


11.50 


2S.27 


11.80 


23.22 


11.70 


26 


27 


24.27 


11.84 


24.22 


11.94 


24.18 


12.05 


24.11 


12.15 


27 


28 


25.17 


12.27 


25.11 


12.38 


25.06 


12.49 


25.00 


12.60 


28 


29 


26.06 


12.71 


26.01 


12.83 


25.95 


12.94 


25.90 


13.05 


29 


30 

ol 


26.98 


13.15 


26.91 


13.27 


26.85 


13.39 
13.83 


26.79 


13.50 


30 

31 


27.86 


13.59 


27.80 


13.71 


27.74 


27.68 


13.95 


32 


28.76 


14.03 


28.70 


14.15 


28.64 


14.28 


28.58 


14.40 


32 


33 


29.66 


14.47 


29.60 


14.60 


29.53 


14.72 


29.47 


14.85 


33 


34 


30.56 


14.90 


30.49 


15.04 


30.43 


15.17 


30.36 


15.30 


34 


35 


31.46 


15.34 


31.39 


15.48 


31.32 


15.62 


31.25 


15.75 


35 


36 


32.36 


15.78 


.32.29 


15.92 


32.22 


18.06 


32.15 


16.20 


36 


37 


33.26 


16.22 


33.18 


16.36 


33.11 


16.51 


33.04 


16.65 


37 


38 


34.15 


16.66 


34.08 


16.81 


34.01 


16.96 


33.93 


17.10 


38 


89 


35.05 


17.10 


34.98 


17.25 


34.90 


17.40 


34.83 


17.55 


39 


40 
41 


35.95 


17.53 


35.87 


17.69 


35.80 


17.85 


35.72 


18.00 


40 

4f 


36.85 


17.97 


36.77 


18.13 


36.69 


18.29 


36.61 


18.45 


42 '37.75 


18.41 


37.67 


18.58 


37.59 


18.74 


37.51 


18.90 


42 


43 


38.65 


18.85 


38.57 


19.02 


38.48 


19.19 


38.40 


19.35 


43 


44 


39.55 


19.29 


39.46 


J9.46 


39.38 


19.63 


39.29 


19.80 


44 


45 


40.45 


19.73 


40.36 


19.90 


40.27 


20.08 


40.18 


20.25 


45 


46 


41.34 


20.17 


41.26 


20.35 


41.17 


20.53 


41.08 


20.70 


46 


47 


42.24 


20.60 


42.15 


20.79 


42.06 


20.97 


41.97 


21.15 


47 


48 


43.14 


21 04 


43.05 


21.23 


42.96 


21.42 


42.86 


21.60 


48 


19 


44.04 


2. .48 


43.95 


21.67 


43.85 


21.86 


43.76 


22.05 


49 


60 

i 

c 

a 


44.94 
Dop. 


21.92 


44.84 


22.11 


44.75 


22.31 


44.65 


22.50 


60 

.1 

Q 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


64 


Deg. 


631 

1 


Deg. 


63^ Deg. 


63t Deg. 



TRAVERSE TABLE. 



126 



3 

s 

a 
? 

"in" 


96Deg. 


m Deg. 


26^ Deg. 


26» 


Deg. 


1 
5J 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 
45.64 


Dep. 

22.76 


Lat, 


Dep. 
32.96 


45.84 


22.36 


45.74 


22.56 


46 84 


62 


46.74 


22.80 


46.64 


23.00 


46.54 


23.20 


46.43 


23.41 


52 


53 


47.64 


23.23 


47.53 


23.44 


47.43 


23.65 


47-33 


23.86 


63 


54 


48.53 


23.67 


48.43 


23.88 


48.33 


24. 09 


48 22 


24.31 


54 


55 


49.43 


24.11 


49.33 


24.33 


49.22 


24.54 


49.11 


24.76 


55 


56 


50.33 


24.55 


50.22 


24.77 


50. J2 


24.09 


50.01 


25.21 


56 


67 


51.23 


24.99 


51.12 


25.21 


51.01 


25 43 


50.90 


25.66 


57 


58 


52.13 


25.43 


52.02 25.65 


51.91 


25 88 


51.79, 


26.11 


58 


69 


53.03 


25.86 


52.92,26.09 


52.80 


26-33 


52.69 


26.-56 


59 


60 
61 


53.93 


26.30 


53.81 


26.54 


53.70 


26-77 

27.22 


53.58 


27.01 


60 

61 


54.83 


26.74 


54.71 


26.98 


54.59 


54.47 


27.46 


62 


55.73 


27.18 


55.61 


27.42 


55.49 


27.66 


55.36 


27.91 


62 


63 


56.62 


27.62 


56.50 


27.86 


56.38 


28.11 


56.26 


28.36 


63 


64 


57 52 


28.06 


57.40 


28.31 


57.28 


28.56 


57.15 


28.81 


64 


65 


58.42 


28.49 


58.30 


28.75 


58.17 


29.00 


58.04 


29.26 


65 


66 


59.32 


28.93 


59.19 


29.19 


59.07 


29.45 


58.94 


29.71 


66 


67 


60.22 


29.37 


60.09 


29.63 


59.96 


29.90 


59.83 


30.16 


67 


68 


61.12 


29.81 


••>0.99| 30.08 1 


60.86 


30.34 


60.72 


30.61 


68 


69 


62.02 


30.25 


61.88 


30.52 


61.75 


30.79 


61.62 


31.06 


69 


70 
71 


62.93 


30.69 


62.78 
63.68 


30.96 


62.65 


31.23 


62.51 


31.51 


70 
71 


63.81 


31.12 


31.40 


63.54 


31.68 


63.40 


31.96 


72 


64.71 


31.56 


64.57 


31.84 


64.44 


32.13 


64.29 


32.41 


72 


73 


65.61 


32.00 


65.47 


32.29 


65.33 


32.57 


65.19 


32.86 


73 


74 


66.51 


32.44 


66.37 


32.73 


66.23 


33.02 


66.08 


33.31 


74 


75 


67.41 


32.88 


67.27 


33.17 


67.12 


33.46 


66.97 


33.76 


75 


76 


68.31 


33.32 


68.16 


33.61 


68.01 


33.91 


67.87 


34.21 


76 


77 


69.21 


33.75 


69.06 


34.06 


68.91 


34.36 


68.76 


34.66 


77 


;8 


70.11 


34.19 


69.96 


34.50 


69.80 


34.80 


69.65 


35.11 


78 


r9 


71.00 


34.63 


70.85 


34.94 


70.70 


35.25 


70.55 


35.56 


79 


SO 

81 


71.90 


.35.07 
35.51 


71.75 


35.38 


71.59 


35.70 


71.44 


36.01 


80 

81 


72.80 


72.65 


35.83 


72.49 


36.14 


72.33 


36.46 


82 


73.70 


35.95 


73.54 


36.27 


73.38 


36.59 


73.22 


36.91 


82 


83 


74.60 


36.38 


74.44 


36.71 


74.28 


37.03 


74.12 


37.36 


83 


84 


75.50 


36.82 


75.34 


37.15 


75.17 


37.48 


75.01 


37.81 


84 


85 


76.40 


37.26 


76.23 


37.59 


76.07 


37.93 


75.90 


38.26 


85 


86 


77.30 


37.70 


77.13 


38.04 


76.96 


38.37 


76.80 


38.71 


86 


87 


78.20 


38.14 


78.03 


38 48 


77.86 


38.82 


77.69 


39.16 


87 


88 


79.09 


38.58 


78.92 


38.92 


78.75 


39.27 


78.58 


39.61 


88 


89 


79.99 


39.01 


79.82 


39.36 


79.65 


39.71 


79.48 


40.06 


89 


90 
91 


80.89 


39.45 


80.72 
81.62 


39.81 
40.25 


80.54 


40.16 


80.37 


40.51 
40.96 


90 
91 


81.79 


39.89 


81.44 


40.60 


81.26 


92 


82.69 


40.33 


82.51 


40.69 


82.33 


41.05 


82.15 


41.41 


92 


93 


83.59 40.77 


83.41 


41.13 


83.23 


41.50 


83.05 


41.86 


93 


94 


84 49 41.21 


84.31 


41.58 


84.12 


41.94 


83.94 


42.31 


94 


95 


85.39 41.65 


85.20 


42.02 


85.02 


42.39 


84.83 


42.76 


95 


96 86.28 42.08 


86.10' 42.46 


85.91 


42.83 


85.73 


43.21 


96 


97 87.18 42.52 


87.00 42.90 


86.81 


43.28 


86.62 


43.66 


97 


98 


88.08 


42.96 


87.89 


43.34 


87.70 


43.73 


87.51 


44.11 


96 


99 


88.98 


43.40 


88.79 


43.79 


88.60 


44.17 


88.40 


44.56 


99 


100 

1 


89.88 


43.84 


89.69 


44.23 


89.49 


44.62 


89.30 


45.01 
Lat. 


100 

• 


Dep. 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


64Dcg. 


63| Deg, 


e^Deg. 


63i Deg. 



126 



rKAVEJt:E TABLE. 



2 
s 

I 

1 


27 Deg. 


27i Deg. 


27^ 


Deg. 


271 Deg. 


ft 

S 

I 


Lat. 


Dep. 
0.45 


Lat 


Dep. 


Lat, 
0.89 


Dep. 

J. 46 


Lat 


Dep. 
0.47 


0.8S 


0.89 


0.46 


0.88 


2 


1.78 


0.91 


1.78 


).i2 


1.77! 


0.92 


1.77 


0.93 


il 


3 


2.67 


1.36 


2.67 


1.37 


2.66 


1.39 


5« 65 


1.40 


3 


4 


3.56 


1 82 


3.56 


1.83 


3.56 


1.85 


'J. 64 1.86 


4 


5 


4.40 


2 27 


4.45 


2.29 


4.44 


2.31 


4.42 2.33 


S 


6 


5.35 


2.72 


5.33 


2.75 


5.32 


2.77 


5.31 


2.79 


fl 


7 


6.24 


3.18 


6.22 


3.21 


6.2] 


3.23 


6.19 


3.26 


7 


8 


7.13 


3.63 


7.11 


3.66 


7.10 


3.69 


7.08 


3.72 


8 


9 


8.02 


4.09 


8.00 


4.12 


7.98 


4.16 


7.96 


4.19 


9 


10 

11 


8.91 


4.54 


8.89 
9.78 


4.58 
5.04 


8.87 


4.62 


8.85 


4.66 


10 

11 


9.80 


4.99 


9.76 


5.08 


9.73 


6.12 


12 


10.69 


5.45 


10.67 


5.49 


10.64 


5.54 


10.62 


5.69 


12 


13 


11.58 


5.90 


11.56 


5.95 


11.53 


6.00 


11.50 


6.05 


13 


14 


12.47 


6.36 


12.45 


5.41 


12.42 


6.46 1 


12.39 


6.52 


14 


15 


13.37 


6.81 


13.34 


6.87 


13.31 


6.93 


13.27 


6.98 


15 


IH 


14.26 


7.26 


14.22 


7.33 


14.19 


7.39 


14.16 


7.45 


16 


17 


15.15 


7.72 


15.11 


7.78 


15.08 


7.85 


16.04 


7.92 


17 


18 


16.04 


8.17 


16.00 


8.24 


15.97 


8.31 


15.93 


8.38 


18 


19 


16.93 


8.63 


16.89 


8.70 


16.85 


8.77 


16.81 


8.86 


19 


20 

21 


17.82 


9.08 


17.78 


9.16 


17.74 


9.23 
9.70 


17.70 


9.31 


20 
21 


18.71 


9.53 


18.67 


9.62 


18.63 


18.58 


9.78 


22 


19.60 


9.99 


19.56 


10.07 


19.51 


10.16 


19.47 


10.24 


22 


23 


20.49 


10.44 


20.45 


10.53 


20.40 


10.62 


20.35 


10.71 


23 


24 


21.38 


10.90 


21.34 


10.99 


21.29 


11.08 


21.24 


11.17 


24 


25 


22.28 


11.35 


22.23 


11.45 


22.18 


11.54 


22.12 


11.64 


26 


26 


23.17 


11.80 


23.11 


11.90 


23.06 


12.01 


23.01 


12.11 


26 


27 


24.06 


12.26 


24.00 


12.36 


23.95 


12.47 


23.89 


12.67 


27 


28 


24.95 


12.71 


24.89 


12.82 


24.84 


12.93 


24.78 


13.04 


28 


29 


25.84 


13.17 


25.78 


13.28 


25.72 


13.39 


26.66 


13.50 


29 


30 
31 


26.73 


13.62 


26.67 


13.74 


26.61 


13.85 


26.65 


13.97 


30 
31 


27.62 


14.07 


27.56 


14.19 


27.50 


14.31 


27.43 


14.43 


32 


28.51 


14.53 


28.45 


14.65 


28.38 


14.78 


28.32 


14.90 


32 


33 


29.40 


14.98 


29.34 


15.11 


29.27 


16.24 


29.20 


15.37 


33 


34 


30.29 


15.44 


30.23 


15.57 


30.16 


15.70 


30.09 


15.83 


84 


35 


31.19 


15.89 


31.12 


16.03 


31.05 


16.16 


30.97 


16.30 


36 


36 


32.08 


16.34 


32.00 


16.48 


31.93 


16.62 


3 '.86 


16.76 


36 


37 


32.97 


16.80 


32.89 


16.94 


32.82 


17.08 


•3i.74 


17.23 


37 


38 


33.86 


17.25 


33.78 


17.40 


33.71 


17.55 


33.63 


17.69 


3S 


39 


34.75 


17.71 


34.67 


17.86 


34.59 


18.01 


34.61 


18.16 


39 


40 

41 


35.64 


18.16 


35.56 


18.31 


35.48 


18.47 


35.40 


18.62 


40 
41 


36.63 


18.61 


36.45 


18.77 


36.37 


18.93 


36.28 


19.09 


42 


37.42 


19.07 


37.34 


19.23 


37.25 


19.39 


37.17 


19.66 


42 


43 


38.31 


19.52 


38.23 


19.69 


38.14 


19.86 


38.05 


20.02 


43 


44 


39.20 


I1& 98 


39.12 


20.15 


39.03 


20.32 


38.94 


20 49 


44 


45 


40.10 


20.43 


40.01 


20.60 


39.92 


20.78 


39.82 


20 95 


45 


46 


40.99 


20.88 


40.89 


21.06 


40.80 


21.24 


40.71 


21.42 


46 


47 


41.88 


21.34 


41.78 


21.52 


41.69 


21.70 


41.69 


21.88 


47 


48 


42.77 


21.79 


42.67 


21.98 


42.58 


22.16 


4^.48 


22.35 


48 


49 


43.66 


22.25 


43.56 


22.44 


43.46 


22.63 


43.36 


22.82 


49 


60 


44.55 
Dep. 


22.70 
1 I^at 


44.45 


22.89 


44.35 


23.03 


4*t.25 


23 28 
Lat. 


50 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 


^ 




























(§ 


63] 


Deg. 


62| Deg. 


6^ 


Deg 


6ai Deg. 


Q 



TRAVERSE TABLE. 



127 



a 
51 


27 Dcg. 


m Dog. 


271 Dog. 


27| Dog. 


Distance, lo 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


15.44 


23.15 


45.34 


23.35 


45.24 


23.55 


45.13 


23.75 


55 


46.33 1 23.61 


46.23 


23.81 


46.12 


24.01 


46.02 


24.21 


52 


53 


47.22 24.06 


47.12 


24.27 


47.01 


24.47 


46.90 


24.68 


53 


54 


48.11 24.52 


48.01 


24.73 


47.90 


24.93 


47.79 


25.14 


51 


56 


49.01 1 24.97 


48.90 


25.18 


48.79 


25.40 


48.67 


25.61 


55 


56 


49.90 1 25.42 


49.78 


25.64 


49.67 


25.86 


49.56 


26.07 


56 


5? 


50.79 


25.88 


50.67 


26.10 


50.56 


26.32 


.50.44 


26.54 57 1 


58 


51.68 


26.33 


51.56 


26.56 


51.45 


26.78 


51.33 


27.01 


58 


59 


52.57 


26 . 79 


52.45 


27.01 


52.33 


27.24 


52.21 


27.47 


59 


60 
61 


53.46 


27.24 


53.34 


27.47 


53.22 


27.70 


53.10 


27.94 


60 
61 


54.35 1 27.69 


54.23 


27.93 


54.11 


28.17 


53.98 


28.40 


62 


55.24 


28.15 


55.12 


28.39 


54.99 


28.63 


54.87 


28.87 


62 


63 


56.13 


28.60 


56.01 


28.85 


55.88 


29.09 


55.75 


29.33 


63 


64 


57.02 


29.06 


56.90 


29.30 


56.77 


29.55 


56.64 


29.80 


64 


65 


57 92 


29.51 


57.79 


29.76 


57.66 


30.01 


57.52 


30.26 


65 


66 58.81 


29.96 


58.68 


30.22 


58.54 


30.48 


58.41 


30.73 


66 


67 


59.70 


30.42 


.59.56 


30.68 


59.43 


30.94 


59.29 


31.20 


67 


68 


60.59 


30.87 


60.45 


31.14 


60.32 


31.40 


60.18 


31.66 


68 


69 


61.48 


31.33 


61.34 


31.59 


61.20 


31.86 


61.06 


32.13 


69 


70 

71 


62.37 


31.78 


62.23 


32.05 


62.09 


32.32 


61.95 


32.59 
33.06 


70 

71 


63.26 


32.23 


63.12 


32.51 


62.98 


32.78 


62.83 


72 


64.15 


32.69 


64.01 


32.97 


63.86 


33.25 


63.72 


33.52 


72 


73 


65.04 


33.14 


64.90 


33.42 


64.75 


33.71 


64.60 


33.99 


73 


74 


65 . 93 


33.60 


65.79 


33.88 


65.64 


34.17 


65.49 


34.46 


74 


75 


66.83 


34.05 


66.68 


34.34 


66.53 


34.63 


66.37 


34.92 


75 


76 


67.72 


34.50 


67.57 


34.80 


67.41 


35.09 


67.26 


35.39 


76 


77 


68.61 


34.96 


68.45 


35.26 


68.30 


35.55 


68.14 


35.85 


77 


78 


69.50 


35.41 


69.34 


35.71 


69.19 


36.02 


69.03 


36.32 


78 


79 


70.39 


35.87 


70.23 


36.17 


70.07 


36.48 


69.91 


36.78 


79 


80 
81 


71.28 


36.32 


71-12 
72.01 


36.63 


70.96 

71.85 


36.94 


70.80 
71.68 


37.25 


80 
81 


72.17 


36.77 


37.09 


37.40 


37.71 


82 


73.06 


37.23 


72.90 


37 . 55 


72.73 


37.86 


72.57 


38.18 


82 


83 


73.95 


37.68 


73.79 


38.00 


73.62 


38.33 


73.45 


38.65 


83 


84 


74.84 


38.14 


74.68 


38.46 


74.51 


38.79 


74.34 


39.11 


84 


85 


75.74 


38.59 


75 . 57 


38.92 


75.40 


39.25 


75.22 


39.58 


85 


86 


76.63 


39.04 


76.46 


39.38 


76.28 


39.71 


76.11 


40.04 


86 


S7 


77.52 


39.50 


77.34 


39.83 


77.17 


40.17 


76.99 


40.51 


87 


88 


78.41 


39.95 


78.23 


40,29 


78.06 


40.63 


77,88 


40.97 


88 


89 


79.30 


40.41 


79.12 


40.75 


78.94 


41.10 


78.76 


41.44 


89 


90 
91 


80.19 


10.86 


80.01 


41.21 


79.83 


41.56 
42.02 


79.65 


41.91 
42.37 


90 
"91 


81.08 


41.31 


80.90 


41.67 


80.72 


80.53 


92 


81 97 


41.77 


81.79 


42.12 


81.60 


42.48 


81.42 


42.84 


92 


93 82.86 


42.22 


82.68 


42.58 


82.49 


42.94 


82.30 


43.30 1 93 1 


94 


83,75 


42.68 


83.57 


43.04 


83.38 43.40 


83.19 


43.77 


94 1 


95 1 


84.65 


43.13 


84.46 


43.50 


84.27 


43.87 


84.07 


44.23 95 1 


96 ' 


85 54 


43.58 


85.35 


43.96 


85.15 


44.33 


84.96 


44.70 96 1 


97' 


80.43 


44.04 


86.23 


44.41 


86.04 


44.79' 


85.84 


45.16 


97 


98 


87.32 


44.49 


87.12 


44.87 


86.93 


45.25 


86.73 


45.63 


98 


99 i 


88.21 


44.95 


88.01 


45.33 


87.81 


45.71 


87.61 


46.10 


99 


100 


89.10 


45^40 
Lat. 


88.90 


45.79 


88.70 


46.17 
Lat. 


88.50 


46.56 


100 

1 


s 1 


Dep. 


Dep. 


Lat. 


Dep. 


Dep. 


Lat. 


63 I 


)eg. 


621 Deg. 


62^ Deg. 


62i Deg. 



128 



TRAVERSE TABLE. 



5' 

P 

a 
p 


28Deg. 


28i Deg. 


36- 


Deg. 


281 Deg. 



B- 

» 
S 

s 

1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


1 Dep. 


1 


0.88 


0.47 


0.88 


0.47 


0.88 


0.48 


0.88 


' 0.48 


2 


1.77 


0.94 


1.76 


0.95 


1.76 


0.95 


1.75 


0.96 


2 


3| 2.65 


1.41 


2.64 


1.42 


2.64 


1.43 


2.63 


1.44 


3 


4 3.53 


1.88 


3.52 


1.89 


3.52 


1.91 


3.61 


1.92 


4 


6 4.41 


2.35 


4.40 


2.37 


4.39 


2.39 


4.38 


2.40 


5 


e 5.30 


2.82 


5.29 


2.84 


5.27 


2.86 


5.26 


2.89 


6 


7 6.18 


3.29 


6.17 


3.31 


6.15 


3.34 


6.14 


3.37 


7 


8 


7.06 


3.76 


7.05 


3.79 


7.03 


3.82 


7.01 


3.86 


8 


9 


7.95 


4.23 


7.93 


4.26 


7.91 


4.29 


7.89 


4.33 


9 


10 
11 


8.83 


4.69 


8.81 


4.73 


8.79 


4.77 


8.77 


4.81 


10 
" 11 


9.71 


5.16 


9.69 


5.21 


9.67 


5.25 


9.64 


5.29 


12 


10.60 


5.63 


10.57 


5.68 


10.55 


5.73 


10.52 


5.77 


12 


13 


11.48 


6.10 


11.45 


6.15 


11.42 


6.20 1 11.40 


6.25 


13 


14 


12.36 


6.67 


12.33 


6.63 


12.30 


6.68 \ 12.27 


6.73 


14 


15 


13.24 


7.04 


13.21 


7.10 


13.18 


7.16! 13.15 


7.21 


15 


16 


14.13 


7.51 


14.09 


7.57 


14.06 


7.63 14.03 


7.70 


16 


17 


15.01 


7.98 


14.98 


8.05 


14.94 


8.11 ! 14.90 


8.18 


17 


18 


15.89 


8.45 


15.86 


8.52 


15.82 


8.59 15.78 


8.66 


18 


19 


16.78 


8.92 


16.74 


8.99 


16.70 


9.07 16.66 


9.14 


19 


20 

21 


17.66 
18.54 


9.39 


17.62 
18.50 


9.47 


17.58 


9.54 
10.02 


17.53 


9.62 


20 
21 


9.86 


9.94 


18.46 


18.41 


10.10 


22 


19.42 


10.33 


19.38 


10.41 


19.33 


10.60 19.29 


10.68 


22 


23 


20.31 


10.80 


20.26 


10.89 


20.21 


10.97 20.16 


11.06 


23 


24 


21.19 


11.27 


121.14 


11.36 


21.09 


11.45 


21.04 


11.64 


24 


25 


22.07 


11.74 


22.02 


11.83 


21.97 


11.93 


21.92 


12.02 


26 


26 


22.96 


12.21 


22.90 


12.31 


22.85 


12.41 


22.79 


12.61 


26 


27 


23.84 


12.68 


23.78 


12.78 


23.73 


12.88 


23.67 


12.99 


27 


28 


24.72 


13.15 


24.66 


13.25 


2-1:. 61 


13.36, 


24.55 


13.47 


28 


29 


25.61 


13.61 


25.55 


13.73 


25.49 


13.84 


25.43 


13.95 


29 


30 


26.49 


14.08 


26.43 


14.20 


26.36 
27.24 


14.31 

14 79 


26.30 


14.43 


30 
31 


31 


27.37 


14.55 


27.31 


14.67 


27.18 


14.91 


32 


28.25 


15.02 


28.19 


15.15 


28.12 


15.i7i 


28.06 


15.39 


32 


33 


29.14 


15.49 


29.07 


15.62 


29.00 


\b.ll 1 


28.93 


16.87 


33 


34 


30.02 


15.96 


29.95 


16.09 


29.88 


16.22' 


29.81 


16.35 


34 


35 


30.90 


16.43 


130.83 


16.57 


30.76 


16.70 


30.69 


16.83 


36 


36 


31.79 


16.90 


131.71 


17.04. 


31.64 


17.18 


31.56 


17.32 


36 


37 


32.67 


17.37 


: 32.59 


17.51 


32.52 


17.65 


32.44 


17.80 


37 


38 


33.55 


17.84 


33.47 


17.99 


33.39 


18.13 


33.32 


18.28 


38 


39 


34.43 


18.31 


! 34.35 


18.46 


34.27 


18.61 


34.19 


18 76 


39 


40 
41 


35.32 


18.78 


35.24 
36.12 


18.93 


35.15 


19.09 
19.56 


35.07 
35.95 


19.24 


40 
41 


36.20 


19,25 


19.41 


36.06 


19.72 


42 


37.08 


19.72 


37.00 


19.88 


36.91 


20.04 


36.82 


20.20 


42 


43 '37.97 


20.19 


137.88 


20.35 


37.79 


20.52 


37.70 


20.68 


43 


44 38 S5 


20.66 


138.76 


20.83 


38 67 


20.99 


38.58 


21.16 


44 


45 39 73 


21.13 


39.64 


21.30 


39.55 


21.47 


39.45 


21.64 


it 


4? 40.62 


21.60 


40.52 


21.77 


40.43 


21.95 


40.33 


22.13 


16 


47 41.50 


22.07 


41.40 


22.25 


41.30 


22.43 


41.21 


22.61 


47 


4^5 42.38 


22.53 42.28 


22.72 


42.18 


22.90 


42.08 


23.09 


48 


49 '43.26 


23.00 43.16 


23.19 


43.06 


23.38 


42.96 


23.57 


49 


50 

i 


44.15 


23.47 


44.04 


23.67 


43.94 


23.86 


43.84 


24.05 


50 

d 


I>ep. 


L&t. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


1 
















c 
















.s 


62 1 


)eg. 6lf Deg. 


64] 


Oeg. 

1 


m 


Deg. 


s 



TRAVERSE TABLK. 



129 



5 

a 
S 

• 

'ft' 


28Deg. 


88iDog. 


28i Deg. 


m Deg. 


s* 

? 

"51 


Lat. 


Dop. 


Lat. 


Dep. 


Lat. 


Dop. 


Lat. 


Dep. 


45.03 


23.94 


44.93 


24.14 


44.82 


24.34 


44.71 


24.53 


52 1 


45.91 


24.41 


45.81 


24.61 


45.70 


24.81 


45.59 


25.01 


^3 


63 


46.80 


24.88 1 


46.69 


25.09 


46.58 


25.29 


46.47 


25.49 


53 


64 


47.68 


25.35 1 


47.57 


25.56 


47.46 


25.77 


47.34 


25.97 


54 


55 


48.56 


25.82 


48.45 


26.03 


48.33 


26.24 


48.22 


26.45 


55 


66 


49.45 


26.29 


49.33 


26.51 


49.21 


26.72 


49.10 


26.94 


56 


57 


50.33 


26.76 


50.21 


26.98 


50.09 


27.20 


49.97 


27.42 


57 


58 


51.21 


27.23 


51.09 


27.45 


50.97 


27.68 


50.85 


27.90 


58 


59 


52.09 


27.70 1 


51.97 


27.93 


51.85 


28.15 


51.73 


28.38 


59 


60 
61 


52.98 


28.17 


52.85 


28.40 


52.73 


28.63 


52.60 


28.86 


60 
61 


53.86 


28.64 


53.73 


28.87 


53.61 


29.11 


53.48 


29.34 


62 


54.74 


29.11 


54.62 


29.35 


54.49 


29.58 


54.36 


29.82 


62 


63 


55.63 


29.58 


55.50 


29.82 


55.37 


30.06 


55.23 


30.30 


63 


64 


56.51 


30.05 


56.38 


30.29 


56.24 


30.54 


56.11 


30.78 


64 


65 


57.39 


30.52 


.57.26 


30.77 


57.12 


31.02 


56.99 


31.26 


65 


66 


58.27 


30.99 


58.14 


31.24 


58.00 


31.49 


57.86 


31.75 


66 


67 


59.16 


31.45 


59.02 


31.71 


58.88 


31.97 


58.74 


32.23 


67 


68 


60.04 


31.92 


59.90 


32.19 


59.76 


32.45 


59.62 


32.71 


68 


69 


60.92 


32.39 


60.78 


32.60 


60.64 


32.92 


60.49 


33.19 


69 


70 
71 


61.81 


32.86 
33.33 


61.66 
62.54 


33.13 


61.52 
62.40 


33.40 


61.37 


33.67 


70 

71 


62.69 


33.61 


33.88 


62.25 


34.15 


72 


63.57 


33.80 


63.42 


34.08 


63.27 


34.36 


63.12 


34.63 


72 


73 


64.46 


34.27 


64.30 


34.55 


64.15 


34.83 


64.00 


35.11 


73 


74 


65.34 


34.74 


65.19 


35.03 


65.03 


35.31 


64.88 


35.59 


74 


75 


66.22 


35.21 


66.07 


35.50 


65.91 


35.79 


65.75 


36.07 


75 


76 


67.10 


35.68 


66.95 


35.97 


66.79 


36.26 


66.63 


36.56 


76 


77 


67.99 


36.15 


67.83 


36.45 


67.67 


36.74 


67.51 


37.04 


77 


78 


68.87 


36.62 


68.71 


36.92 


68.. 55 


37.22 


68.38 


37.52 


78 


79 


69.75 


37.09 


69.59 


37.39 


69.43 


37.70 


69.26 


38.00 


79 


80 
81 


70.64 


37.56 


70.47 


?7.87 


70.31 
71.18 


38.17 
38.66 


70.14 


38.48 
38.96 


80 


71.52 


38.03 


71.35 


38.34 


71.01 


81 


82 


72.40 


38.50 


72.23 


38.81 


72.06 


39.13 


71.89 


39.44 


82 


83 


73.28 


38.97 


73.11 


39.29 


72.94 


39.60 


72.77 


39.92 


83 


84 


74.17 


39.44 


73.99 


39.76 


73.82 


40.08 


73.64 


40.40 


84 


85 


75.05 


39.91 


74.88 


40.23 


74.70 


40.56 


74.52 


40.88 


85 


86 


75.93 


40.37 


75.76 


40.71 


75 58 


41.04 


75.40 


41.36 


86^ 


87 


76.82 


40.84 


76.64 


41.18 


76.46 


41.51 


76.28 


41.85 


87 


88 


77.70 


41.31 


77.52 


41.65 


77.34 


41.99 


77.15 


42.33 


88 


89 


78.58 


41.78 


78.40 


42.13 


78.21 


42.47 


78.03 


42.81 


89 


90 

91 


79.47 


42.25 


79.28 


42.60 


79.09 


42.94 


78.91 


43.29 


90 
91 


80.35 


42.72 


80.16 


43.07 


79.97 


43.42 


79. /S 


43.77 


92 


81.23 


43.19 


81.04 


43.55 


80.85 


143.90 


80.66 


44 25 


92 


93 


82.11 


43.66 


81.92 


44.02 


81.73 


44.38 


81.54 


44 73 


93 


94 


83.00 


44.13 


82.80 


44.49 


82.61 


44.85 


82.41 


^.fl 


91 


95 


83.88 


44.60 


83.68 


44.97 


83.49 


45.33 


83.29 


45. u9 


95 


96 


84.78 


45.07 


84.57 


45.44 


84.37 45.81 


84.17 


146.17 


96 


97 


85.65 


45.54 


85.45 


45.91 


85.25 


46.28 


85.04 


1 46.66 


97 


98 


86.53 


46.01 


86 33 


46.39 


86.12 


46.76 


85.92 


47.14 


98 


99 


87.41 


46.48 


87.21 


46.86 


87.00 


47.24 


86.80 


47.62 


99 


100 

— r 


88.29 


46.95 


88.09 


47.33 


87.88 


47.72 


87.67 
Dep. 


48.10 


too 

§ 

1 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 Lat. 


1 Lat. 


62 Deg. 


611 Deg. 


61i Deg. 


6U Deg. 



130 



TRAVERSE TABLE. 



a 

S' 

§ 
S 


29 Deg. 


29i Deg. 


29i Deg. 


29i Deg. 


C 
5* 

s 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 1 


Dep. 


Lat. j 


Dep. 


■ " 0.87 1 


48 


0.87 : 


0.49 i 


0.87 


0.49 


0.87 


0.50 


1 


2 1.75 1 


0.97 


1.74 1 


0.98 


1.74 


0.98 


1.74 


0.99 


\ 


3 2.62 1 


1.45 


2.62 1 


1.47 ; 


2.61 


1.48 


2.60 


1.49 3 


4 I 3.50 1 


1.94 I 


3.49 1 


1.95 


3.48 


1.97 : 


3.47 


1.98 4 


6 4 37| 


2.42 


4. 36 


2.44 


4.35 


2.46 ' 


4.34 


2.48 5 


6 


5.25 


2.91 


5.23 


2.93 


5.22 


2.95 1 


5.21 


2.98 6 


7 


6.12 


3.39 


6.11 


3.42 


6.09 


3.45 


6.08 


3.47! 7 


8 7.00 1 


3.88 


6.98 


3.91 


6.96 


3.94 


6.95 


3.97 8 


9 


7.87 1 


4.36; 


7.85 


4.40 


7.83 


4.43 


7.81 1 


4.47 ^ 9 


10 
11 


8.75 


4.85; 


8.72 
9.60 


4.89 


8.70 


4,92 


8.68 1 
9.55 i 


4.96 10 
5,46 11 


9.62 


5.33 


5.37 


9.57 


5.42 


12 


10.50 


5.82 


10.47 


5.86 


10.44 


5.91 


10.42 


5.95 12 


13 


11.37 


6.30 


11.34 


6.35 


11.31 


6.40 


11.29, 


6.45 13 


14 


12.24 


6.79 


12.21 


6.84 


12.-18 


6.89 


12.15! 


6.95 14 


15 


13.12 


7.27 


13.09 


7.33 


13.06 


7.39 


13.02 


7.44 15 


16 


13.99 


7.76 


13.96 


7.82 


13.93 


7.88 


13.89 


7.94 16 


17 


14.87 


8.24 


14.83 


8.31 


14.80 


8.37 


14.76 


8.44 17 


18 


15.74 


8.73 


15.70 


8.80 


15.67 


8.86 


15.63 


8.93 18 


19 


16.62 


9.21 


16.58 


9.28 


16.. 54 


9.36 


16.50 


9.43 19 


20 

21 


17.49 
18.37 


9.70 
10.18 


17.45 
18.32 


9.77 
10.26 


17.41 

18.23 


9.85 
10.34 


17.30 


9.92 ! 


20 
21 


18.23 


10.42 1 


22 1 19.24 


10.67 


19.19 


10.75 


19.15 


10.83 


19.10 


10.92 


22 


23 20.12 


11.15 


20.07 


11.24 1 


20 . 02 


11.33 


19.97 


11.41 


2« 


24 ,20.99 


11.64 


20.94 


11.73 


20.89 


11.82 


20.84 


11.91 


24 


25 ,21.87 


12.12 


21.81 


12.22 


21.76 


12.31 


21.70 


12.41 ! 


25 


26 


22.74 


12.60 


22.68 


12.70 


22.63 


12.80 


22.57 


12.90 


26 


27 


23.61 


13.09 


23.56 


13.19 


23.50 


13.30 


23.44 


13.40 


27 


28 


24.49 


13.57 


24.43 


13.68 


24.-37 


13.79 i 


24.31 


13.89 1 


28 


29 


25.36 


14.06 


25.30 


14.17 


25.24 


14.28 ■ 


25.18 


14.39 I 


29 


30 


20.24 


14.54 


26.17 


1 14.66 
i 15.15 


26.11 


14.77 


26.05 
26.91 


14.89 
15.38 


30 
31 


31 


27.11 


15.03 


27.05 


26.98 


15.27 


32 


27.99 


15.51 


27.92 


1 15.64 


27.85 


15.76 


27.78 


15.88 


32 


33 


28.86 


16,00 


28.79 


16.12 


28.72 


16.25 


28.65 


16.38 


33 


34 1 29 . 74 


16.48!, 29.66 


1 16.61 


29.59 


16.74 


(29.52 


16.87 ■ 34 


35 130.61 


16.97 


30.54 


17.10 


30.46 


17.23 


30.39 


17.37 35 


36 


31.49 


17.45 


31.41 


17.59 


31.33 


17.73 


' 31.26 


17.86 36 


37 


32.36 


! 17.94 


32.28 


18.08 


32.20 


18.22 


1 32.12 


18.36 


37 


38 


33.24 


i 18.42 


33.15 


18.57 


33.07 


18.71 


i 32.99 


18.86 


38 


39 


34.11 


! 18.91 


34.03 


19.06 


33.94 


19.20 


33.86 


19.35 


39 


40 


34.98 


! 19.39 


34.90 
35 . 77 


19.54 


34.81 
35.68 


19.70 
20.19 


34.73 
35.60 


19.85 
20.34 


40 
41 


41 


135.86 


19.88 


20.03 


42 ' 36 . 73 


20.36 


36.64 


20.52 


36.55 


20.68 


1 36.46 


1 20.84 


42 


43 37.61 


20.85" 37.52 


21.01 


137.43 


21.17 


37.33 


21.34 


43 


44 38.48 


21.33 38.39 


21.50 


1 38.30 


21.67 


! 38.20 


' 21.83 


44 


45 '39.36 


21.82 i 39.26 


21.99 


139.17 


22.16 


39.07 


22 . 33 45 


46 40.23 22.30 1; 40.13 


122.48 


; 40 . 04 


22.65 


39.94 


22.83 46 


47 41.11 


22.79 


141.01 


122.97 


! 40.91 


23.14 


40.81 


23.32 


47 


4« '41.98 


23.27 


I 41.88 


23.45 


41.78 


23.68 


i 41.67 


' 23.82 


1 48 


49 1 42 86 


23.76 


; 42.75 


23.94 


42.65 


24.13 


1 42.54 


i 24.31 


' 49 


50 Ii3.73 


24.24 


43.62 


24.43 


43.52 


24.62 
Lat. 


43.41 


! 24.81 


50 


i 


Dep. 


1 Lat. 


Dep. 


Lat 


Dop. 


Dep. 


1 Lat. 


o 

a 


61 Dcg. 

1 


601 


Deg. 


60J Deg. 


6(H Deg. 





TRAVERSE TAKLE. 



131 



p . 

P 
51 


29 Deg. 


29i Deg. 


29i Deg. 


29| Deg. 


5 

5' 

C3 
a 
p 


Lat. 

44.61 


Dep. 
24.73 


Lat. 


Dep. 


Lat. 


Dep. 

25.11 


Lat. 


Dep. 

25731 


44.50 


24.92 


44.39 


44.28 


52 


45.48 


25.21 


45.37 


25.41 


45.26 


25.61 


45.15 


25.80 


52 


63 


46.35 


25.69 


46.24 


25.90 


46.13 


26.10 


46.01 


26.30 


63 


r>4 


47.23 


26.18 


47.11 1 


26.39 


47.00 


26.59 


46.88 


26.80 


54 


65 


48.10 


26.66 


47.99 : 26.87 | 


47.87 


27.08 


47.75 


27.29 


55 


56 


48.98 


27.15 1 


48.86 


27.36 


48.74 


27.58 


48.62 


27.79 


66 ' 


57 


49.85 


27.63! 


49.73 


27.85 


49.61 


28.07 


49.49 


28.28 


57 


bS 


50.73 


28.12 


50.60 


28.34 


50.48 


28.56 


50.36 


28.78 


58 


59 


51.60 


28.60 


51.48 


28.83 


51.35 


29.05 


51.22 


29.28 


69 


60 
61 


52.48 
53.35' 


29.09 
29.57 


52.35 


29.32 


52.22 
53.09 


29.55 
30.04 


52.09 
52.96 


29.77 
30.27 


60 
61 


53.22 


29.81 


62 


54.23 


30.06 


54.09 


30.29 


53.96 


30.53 


53.83 


30.77 


62 


63 


55.10 


30 . 54 


54.97 


30.78 


54.83 1 31.02 1 


54.70 


31.26 


63 


64 


55.98 


31.03 


55.84 


31.27 


55.70 


31.52 


55.56 


31.76 


64 


65 


i,t>.85 


31 51 


56.71 


31.76 


56.57 


32.01 


56.43 


32.25 


65 


66 


^7.72 


32 . 00 


57.58 


32.25 


57.44 


32.50 


57.30 


32.75 


66 


67 


.58.00 


32.48 


.58.40 


32.74 


.58.31 


32.99 


58.17 


33.25 


67 


68 


59.47 


32.97 


59.33 


33.23 


59.18 


33.48 


59.04 


33.74 


68 


69 


60.35 


33.45 


60.20 


33.71 


60.05 


33.98 1 


59.91 


34.24 


69 


70 
71 


61.22 
62.10 


33.94 

34.42 


61.07 


34.20 
34.69 


60.92 
61.80 


.34.47 
34.96 


60.77 


34.74 


70 
71 


61.95 


61.64 


35.23 


72 


62.97 


34.91 


62.82 


35.18 


62.67 


35.45 ! 


62.51 


35.73 72 


73 


63.85 


35.39 


63.69 


35.67 


63.54 


35.95 


63.38 


36.22 73 


74 


64.72 


35.88 


64.56 


36.16 


64.41 


36.44 


64.25 


36.72 


74 


75 


65 . 60 


36.36 


65.44 


36.65 


65.28 


36.93 


65.11 


37.22 


75 


76 


66.47 


36.85 


66.31 


37.14 


66.15 


37.42 


65.98 


37.71 


76 


77 


67.36 


37.33 


67.18 


37.62 


67.02 


37.92 


66.85 


38.21 


77 


78 


68.22 


37.82 


68.05 


38.11 


67.89 


38.41 


67.72 


38.70 


78 


79 


69.09 


38.30 


68.93 


38.60 


68.76 


38.90 


68.59 


39.20 


79 


80 
81 


69.97 


38.78 


69.80 


39.09 
39.58 


69.63 


39.39 
39.89 


69.46 


39.70 


80 
81 


70.84 


39.27 


70.67 


70.50 


70.32 


40.19 


82 


71.72 


39.75 


71.54 


40.07 


71.37 


40.38 


71.19 


40.69 


82 


83 


72.59 


40.24 


72.42 


40.56 


72.24 


40.87 


72.06 


41.19 


83 


84 


73.47 


40.72 


73.29 


41.04 


73.11 


41.36 


72.93 


41.68 


84 


85 


74.34 


41.21 


74.16 


41.53 


73.98 


41.86 


73.80 


42.18 


85 


86 


75 . 22 


41.69 


j 75.03 


42.02 


74.85 


42.35 


74.67 


42.67 


86 


87 


76.09 


42.18 


75.91 


42.51 


75.72 


42.84 


75.53 


43.17 


1 87 


88 


76.97 


42.66 


76.78 


43.00 


76.59 


43.33 


76.40 


43.67 


88 


89 


77.84 


43.15 


77 . 65 


43.49 


77.46 


43.83 


77.27 


44.16 


89 


90 
91 


78.72 


43.63 


78 . 52 
79.40 


43.98 
44.46 


78.33 


44.32 

44.81 


78.14 


44.66 


90 1 


79.59 44.12 


79.20 


79.01 


45.16 


91 


92 80.46 44.60 


80.27 


44.95 


80.07 


45.30 


79.87 


45.65 95 


93 81.34 45.09 


81.14 


45.44 


SO. 94 


45.80 


80.74 146.15 09 


94 82.21 45.57 


82.01 


45.93 


81. SI 


46.29 


81.61 


46.64 91, 
47.14 951 


95 


83.09 46.06 


82.89 


46.42 


82.68 


46.78 


82.48 


96 


89.96 46.54 


83.76 


46.91 


83.55 


47.27 


83.35 


1:7,64 


1 96 


97 


84.84 47.03 


84.63 


47.40 


84.42 


47.77 


84.22 


48 13 


97 


98 


85.71 


t 47.51 


85.50 


47.88 


85.29 


48.26 


85.08 


48.63 


98 


99 


86.59 


48.00 


86.38 


48.37 


86.17 


48.75 


85.95 


49.13 


99 


iOO 

s 


87.46 
Dep. 


48.48 


87.25 


48.86 


87.04 
Dep. 


49.24 


86.82 
Dep. 


49.62 
Lat. 


100 

o 

o 

a 
ed 

1 


Lat. 


Dep. 


Lat. 


Lat. 


il 


61 


Dt^. 


601 Deg. 


601 Deg. 


60^ Deg. 



132 



TRAVERSE TADIB. 



s* 

s 

• 

I 


SODeg. 


SOiDeg. 


SOf 


Deg. 


301 Deg. 


§ 

s 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 
0.61 


0.87 


0.50 


0.85 


0.50 


3.86 


0.51 


0.86 


1 


% 


1.73 


1.00 


1.73 


1.01 


1.72 


'.02 


1.72 


1.02 


3 


3 


2.60 


1.50 


2 59 


1.51 


2.58 


1.52! 


2 58' 1.53 


3 


4 


3.46 


2.00 


3.46 


2.02 


3.45 


2.03 


3 44 


2.05 


i 


5 


4.33 


2.50 


4.32 


2.52 


4.31 


2.54 


4 30 


2.56 


fi 


6 


5.20 


3.00 


5.18 


3.02 


5.17 


3.05 


5.16 


3.07 


5 


7 


6.06 


3.50 


6.05 


3.53 


6.03 


3.55 


6.02 


3.58 


7 


8 


6.93 


4.00 


6.91 


4.03 i 


6.89 


4.06 11 6.88 


4.09 


8 


9 


7.79 


4.50 


7.77 


4.53 1 


7.75 


4.571 


7.73 


4.60 


9 


lU 
11 


8.66 
9.53 


5.00 


8.64 


5.04 1 


8.62 


5.08 1 


8.59 


5.11 


10 


5.50 


9.50 


5.54 1 


9.48 


5.58 i 


9.45 


5.62 


12 


10.39 


6.00 


10.37 


6.05 


10.34 


6.09' 


10.31 


6.14 


12 


13 


11.26 


6.50 


11.23 


6.55 


11.20 


6.60 


11.17 


6.65 


13 


14 


12.12 


7.00 


12.09 


7.05 


12.06 


7.11 


12.03 


7.16 


14 


15 


12.99 


7.50 


12.96 


7.56 


12.92 


7.61 


12.89 


7.67 


15 


16 


13.86 


8.00 


13.82 


8.06 


13.79 


8.12 


13.75 


8.18 


16 


17 


14.72 


8.50 


14.69 


8.56 


14.65 


8.63 


14.61 


8.69 


17 


18 


15.59 


9.00 


15.55 


9.07 


15.51 


9.14 


15.47 


9.20 


18 


19 


16.45 


9.50 


16.41 


9.57 


16.37 


9.64 


16.33 


9.71 


19 


20 
21 


17.32 


10.00 


17.28 


10.08 ! 


17.23 


10.15 
10.66 


17.19 


10.23 
10.74 


20 

21 


18.19 


10.50 


18.14 


10.58 


18.09 


18.05 


22 


19.05 


11.00 


19.00 


11.08 


18.96 


11.17 


18.91 


11.25 


22 


23 


19.92 


11.50 


19.87 


11.59 


19.82 


11.67 


19.77 


11.76 


23 


24 


20.78 


12.00 


20.73 


12.09 


20.68 


12.18 


20.63 


12.27 


24 


25 


21.65 


12.50 


21.60 


12.59 


21.54 


12.69 


21.49 


12.78 


25 


26 


22.52 


13.00 


22.46 


13.10 


22.40 


13.20 


22.34 


13.29 


26 


27 


23.38 


13.50 


23.32 


13.60 


23.26 


13.70 


23.20 


13.80 


27 


28 


24.25 


14.00 


24.19 


14.11 


24.13 


14.21 


24.06 


14.32 


28 


29 


25.11 


14.50 


25.05 


14.61 


24.99 


14.72 


24.92 


14.83 


29 


30 
31 


25.98 


15.00 


25.92 


15.11 


25.85 


15.23 


25.78 


15.34 


30 
31 


26.85 


15.50 


26.78 


15.62 


26.71 


15.73 


26.64 


15.85 


32 


27.71 


16.00 


27.64 


16.12 


27.57 


16.24 


27.50 


16.36 


32 


33 


28.58 


16.50 


28.51 


16.62 


28.43 


16.75 


28.36 


16.87 


33 


34 


29.44 


17.00 


29.37 


17.13 


29.30 


17.26 


29.22 


17.38 


34 


35 


30.31 


17.50 


30.23 


17.63 


30.16 


17.76 


30.08 


17.90 


35 


36 


31.18 


18.00 


31.10 


18.14 


31.02 


18.27 


30.94 


18.41 


36 


37 


32.04 


18.50 


31.96 


18.64 


31.88 


18.78 


31.80 


18.92 


37 


38 


32.91 


19.00 


32.83 


19.14 


32.74 


19.29 


32.66 


19.43 


38 


39 


33.77 


19.50 


33.69 


19.65 


33.60 


19.79 


33.52 


19.94 


39 


40 

41 


34.64 


20.00 


34.55 


20.15 


34.47 


20.30 


34.38 


20.45 


40 
'41 


35.51 


20.50 


35.42 


20.65 


35., 33 


20.81 


35.24 


20.96 


42 


36.37 


21.00 


136.28 


21.16 


36.19 


21.32 


36,10 


21.47 


42 


43 


37.24 


21.50 


!37 14 


21.66 


37.05 


21.82 


36.95 


21.99 


43 


44 


38.11 


22.00 


38.01 


22.17 


37.91 


22.33 


37.81 


22.50 


44 


45 


38.97 


22.50 


38.87 


22.67 


38.77 


22.84 


38.67 123.01 


15 


46 


39.84 


23.00 


39.74 


23. ]7 


39.63 


23.35 


39.53 


23.52 


46 


4r 40.70 


23.50 


40.60 


23.68 


40.50 


23.85 


40.39 


24.03 


4? 


48,41.57 


24.00 


41.46 


24.18 


41 36 


24.36 


41.25 


24.54 


48 


49 


42.44 


24.50 


42.33 


24.68 


42.22 


24.87 


42.11 


25.05 


4S 


60 

s 

.a 


43.30 


25.00 


43.19 


25.19 


43.08 


125.38 


42.97 


25.66 


60 

B 

00 

s 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


60 1 


)eg. 


691 Deg. 


59i 


Deg. 


59i Deg. 



TRAVERSE TABLE. 



133 



9 

r*- 
P 

51 


30Deg. 


30i Deg. 


m Deij. 


30| Deg. 


Q 

B 
a 
o 
a 

51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


44.17 


25.50 


44.06 


25.69 


43.94 


25.88 


43.83 


26.08 


52 


45.03 


26.00 


44.92 


26.20 


44.80 


26.39 


44.69 


26.69' 52 


53 ' 45.90 


26 50 


45.78 


26.70 


45.67 


26.90 


45.56 


27.10 


5,3 


54 46.77 


27 00 


46.65 


27.20 


46.53 


27.41 


46.41 


27.61 


64 


65 


47.63 


27.50 


47.51 


27.71 


47.39 


27.91 


47.27 


28.12 


65 


56 


48.60 


28.00 


48.37 


28.21 


48.26 


28.42 


48.13 


28.63 


56 


57 


49.36 


28.50 


49.24 


28.7:> 


49.11 


28 98 


48.99 '29.14 


67 


68 


50 23 


29.00 


50.10 


29. 2i 


149.97 


29.44 


49.85 


29.65 


58 


69 


51.10 


29.50 


50.97 


29.72 


150.84 


29.94 


50.70 


30.17 


59 


60 

61 


51.96 
52.83 


3U.00 


51.83 


30.23 


61.70 


30.45 


51.56 


30.68 


60 

61" 


30.50 


52.69 


30.73 


52.. 56 


30.96 


52.42 


31.19 


62 


53.69 


31.00 


53.56 


31.23 


53.42 


31.47 


63.28 


31.70 


62 


63 


54.56 


31.50 


54.42 


31.74 


54.28 


31.97 


64.14 


32.21 


63 


64 


55.43 


32.00 


55.29 


.32.24 


55.14 


32.48 


55.00 


32.72 


04 


65 


56.29 


32.50 


56.15 


32.75 


56.01 


32.99 


55.86 33.23 


65 


66 


57.16 


33.00 


57.01 


33.26 


.56.87 


33.50 


.56.72 


33.75 


66 


67 


58.02 


33.50 


57.88 


33.75 


t>7.73 


34.01 


57.58 


34.26 


67 


68 


58.89 


34.00 


68.74 


34.26 


58.59 


34.61 


58.44 


34.77 


68 


69 


59.76 


34.60 


59.60 


34.76 


59.45 


35.02 


69.30 


35.28 


69 


70 
71 


60.62 


35.00 


60.47 


35.26 
35.77 


60.31 


35.53 


60.16 
61.02 


35.79 
36.30 


70 

71 


61.49 


35.. 50 


61.33 


61.18 


36.04 


72 


62.35 


36.00 


62.20 


36.27 


62.04 


36.64 


61.88 


36.81 


72 


73 


63.22 


36.50 


63.06 


36.78 


62.90 


37.05 


62.74 


37.32 


73 


74 


64.09 


37.00 


63.92 


37.28 


63.76 


37.56 


63.60 


37.84 


74 


75 


64.95 


37.50 


64.79 


37.78 


64.62 


38.07 


64.46 


38.35 


75 


76 


65.82 


38.00 


65.66 


38.29 


65.48 


38.57 


65.31 


38.86 


76 


77 


66.68 


38.50 


66.52 


38.79 


66.35 


39.08 


66.17 


39.37 


77 


78 


67.55 


39.00 


67.38 


39.29 


67.21 


39.69 


67.03 


39.88 


78 


79 


68.42 


39.. 50 


68.24 


39.80 


68.07 


40.10 


67.89 


40.39 


79 


80 
81 


69.28 


40.00 


69.11 
69.97 


40.30 


68.93 


40.60 


68.76 


40.90 


80 


70.15 


40.50 


40.81 


69.79 


41.11 


69.61 


41.41 


81 


82 


71.01 


41.00 


70.83 


41.31 


70.65 


41.62 


70.47 


41.93 


82 


83 


71.88 


41.. 50 


71.70 


41.81 


71.52 


42.13 


71.33 


42.44 


83 


84 


72.75 


42.00 


72.66 


42.. 32 


72.38 


42.63 


72.19 


42.96 


84 


85 


73.61 


42.50 


73.43 


42.82 


73.24 


43.14 


73.06 


43.46 


86 


86 


74.48 


43.00 


74.29 


43.32 


74.10 


43.66 


73.91 


43.97 


86 


87 


75.34 


43.50 


76.15 


43.83 


74.96 


44.16 


74.77 


44.48 


87 


88 


76.21 


44.00 


76.02 


44.33 


76.82 


44.66 


76.63 


44.99 


88 


89 


77.08 


44.50 


76.88 


44.84 


76.68 


46.17 


76.49 


45.61 


89 


90 
91 


77.94 


45.00 


77.76 


45.34 


77.66 


46.68 


77.35 


46.02 


90 
91 


78.81 


45.50 


78.61 


45.84 


78.41 


46.19 


78.21 


46.63 


92 


79.67 


46.00 


79.47 


46.35 


79.27 


46.69 


79.07 


47.04 


92 


93 


80.64 


46.50 


80.34 


46.85 


80.13 


47.20 


79.93 


47.55 


93 


94 


81.41 


47.00 


81.20 


47.35 '80.99 


47.71 


80.78 


48.06 


94 


95 


82.27 


47.50 


82.06 


47.86 81.86 


48.22 


81.64 


48.57 


95 


96 


83.14 48.00 


82.93 


48.36 82.72 


48.72 


82.50 


49.08 


96 


9? 


84.00 48.50 


83.79 


48.87 83.58 


49.23 


83.36 


49.60 


97 


98 


84.87 


49.00 


84.66 


49.37 84.44 


49.74 


84.22 


60.11 


98 


95 85.74 


49.5fl 


85.62 


49.87 86.30 


50.25 


86.08 


60.62 


99 


too 

§ 

c 
a 

«-» 


86.60 
Dep. 


50.00 


86.38 60.38 


86.16 


60.76 


86.94 


51.13 


100 

O 

c 

s 

■n 

c 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


60 Dog. 


591 Deg. 59i Dog. 


59i Deg. 



134 



TEAVER5E! TABLE 



>^ 


31 ] 


D«g- 


a 












■^ 


Lv., 


-'^1- 


i 


5D 


u..5i 


8 


in 


1.03 


3 


2.57 


1 . 55 


4 


3.43 


2 . L' 5 


5 


i.-2j 


'.; . ? 


6 


5 . 14 


3.:^ 


7 


6 , 'JL' 


3 . -:■ . 


:> 


~ -^ - 


4 _ , 


U 




4,0^ 


lu 


* .0 , 


. .0 


11 


- 43 


5 . oT 


12 


■ 


^. V 


13 


... 14 


'"* 7 ' 



3i\ D«^. 



3U I>«g. 



3U Dej. 



14 
15 
16 



12.00 

12. S6 
13.71 



7.7c 
8.24 



26 
27 
28 

29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

41' 

42 

43 

44 

45 

46 

IS 
49 
50 



■^ 1 


57 


12 


36 


^ ]_ 


43 


12 


«5 


22 


29 


13 


C ^ 


23 


14 


13 


,_- '_ 


24 


00 


14 


-r'J 


24 


86 


14 


y-t 


25 


71 


15 


45 


26 


57 


15 


97 


■^ i 


43 


' P 


-i *^ 


• »* 




:I 





29.14 
30 . 00 
30.86 
31.72 
32.57 
33.43 
^4.29 

35.14 
36.00 
35.86 
37.72 
33 . 57 
39.43 
40.29 
41.14 
42.00 
42.86 



i I 

18.03 

18.. 54 

19.06 

19.57 

20.09 

20 . 60 

21.12 

2l!63 

22.15 

22 . P '^ 

23! 15 

23.-:^ 

24 . ■; : 

24.72 

25 . 24 
25.75 



La: 



Dep. Lat, Dep. Lat. Dep. 



U,8c 
1.71 
2.56 
3.42 



13 



0.52 
1.04 

i.55 
2 .OS 

2.59 

3.11 
■^ . - 3 






85 


0.52 


1 


71 


1.04 


r> 


- ~, 


1.57 




-'_ 


2.09 


_ 


■^ J 


2.61 


5 


■ .-> 


3.13 


5 


97 


3.66 


•3 


82 


4.18 


"" 


67 


4.70 



53 



13, 



53 



30 17.10 10. 3S 



17.95 

1 -^ . SI 

- r . 6 6 

20.52 

21.37 

■:: 23 



25 . 65 



10.89 
11.41 1 
11.93!, 
12.45 
12.97 ! 
13.49 ' 
14.01 
14.53 ' 
15.04 
15.56 



29 


92 


30 


1 5 


31 


63 


32 


49 


33 


34 


34 


20 



26 . 50 1 6 . 08 



18.16 
18.68 
19.19 
19.71 
20 . 23 
20.75 

21.27 
21.79 
22.31 
22.83 
23 . 34 
23.86 
24.38 
24.90 
25.42 
25.94 



35 


05 


35 


91 


36 


/ b 


3- 


62 


:;S 


^~ 


Sr 


33 



41.89 
i2.75 



^.35 

: .23 

.1.08 
11.94 
12.79 
13.64 
14.49 
1 S 35 
: : . 20 



13.76 
19.61 
20.46 
21.32 
22.17 
23.02 
23.87 
24.73 
25.58 
26.43 
27. 2-8 
,-.14 
2S.99 
29.84 
30 . 70 
31.55 
32.40 
33 . 25 
34.11 

34.96 
35. SI 
36.66 
37 . 52 
3S.37 
39.22 
40.07 
4J.93 



5 . 22 

5.75 
6.27 
6.791 
7.31 

7-84 

8.36 

8.88 1 

9.401 

9.93 I 

10.45 

10,97 

11.49 

12.02 

12.54 

13.06 

13.5.3 

14.11 

14.63 1 

15.15 

15.67 

16.20 
16.72 
17.24 
17.76, 
13.29 1 
13.81 1 
19.33 
19.85 
20 . 38 
20 . 90 

21.42 
21.94 
22.47 
22 . 99 
23.51 
24.03 
24.56 
25.03 
25.60 
26.12 



o Dep. I Lat. Dep. Lat. Dep. Lat. 



0.S5 
1.70 
2 . 55 
3.40 
4.25 
5. 10 
5.95 
6.80 
7.65 
3.50 

9.. 35 
10.20 
11.05 
11.90 
12.76 
13.61 
14.46 
15.31 
16.16 
17.01 

17.86 
13.71 
19.56 
20.41 
21.26 
22 . 1 1 
22.96 
23.81 
24.66 
25.51 
26.36 
27.21 
28.06 
23.91 
29.76 
30.61 
31.46 
32.31 
33.16 
34.01 
34.86 
35.71 
.?6 . 57 
37.42 
33 . 27 
39.12 
39.97 
40.82 
41.67 
42.52 

Deo. 



0.53 

1 .05 
1 . 53 
2.10 
2.63 
3.16 
3.68 
4.21 
4.74 
5.26 

5. 79 
6.31 
6.84 
7.37 
7.89 
•i.42 
^.95 
9.47 
10.00 
10.52 

11.05 
11.58 
12.10 
12 . 63 
13.16 
13.68 
14.21 
14.73 
15.26 
15.79 
16.31 
16.84 
17.37 
17.89 
13.42 
13.94 
19.47 
20.00 
20.52 
21 05 

21.57 
22.10 
22 . 63 
23.15 
23.63 
24,21 
24.73 
25.26 
25.78 
26.31 

Lat. 



3 
p 

"T 

2 
3 
4 
5 
6 
7 
S 
9 
10 

11 
12 
13 

14 
15 
16 
17 
18 
19 
20 



23 
24 
25 



29 
_30 

31 
32 
33 
34 
35 
36 
37 
33 
39 
40 

'41 
42 
43 
44 
4.5 
46 
47 
iS 
49 
50 

e 
o 

c 



59 Dey 



58| Dee. 



58i Deg. 



38t nf<7. 



TRAVERSE TABL£ 



135 



9. 

OB 

a 
a 


1 

31 Deg. 


3U Deg. 


31^ Deg. 


31} Deg. 


O 
5' 

a 

? 

"51 


Lat. 


Dop. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


43.72 


26.27 


43.60 


26.46 


43.48 


26.65 


43.37 


26.84 


62 


44.57 


26.78 


44.46 


26.98 


44.34 


27.17 


44.32 


27.36 


62 


63 


45.43 


27.30 


45.31 


27.49 


45.19 


27.69 


45.07 


27.89 


5fl 


64 


46.29 


27.81 


46.17 


28.01 


46.04 


28.21 


45.92 


28.42 


64 


55 


47.14 


28.33 


47.02 


28.53 


46.90 


28.74 


46,77 


28.94 


56 


56 


48.00 


28.84 


47.88 


29.05 


47.75 


29.26 


47.62 


29.47 


66 


57 


48.86 


29.36 


48.73 


29.57 


48.60 


29.78 


48.47 


29.99 


67 


58 


49.72 


29.87 


49 58 


30.09 


49.45 


30.30 


49.32 


30.52 


68 


59 


50.57 


30.39 


50.44 


30.61 


50.31 


30.83 


50.17 


31.05 


69 


60 
61 


51.43 


30.90 


51.29 


31.13 


51.16 


31.35 


51.02 


31.57 


60 
61 


52.29 


31.42 


52.15 


31.65 


52.01 


31.87 


51.87 


32.10 


62 


53.14 


31.93 


53.00 


32.16 


52.86 


32.39 


52.72 


32.63 


62 


63 


54.00 


32.45 


53.86 


32.68 


53.72 


32.92 


53.57 


33.16 


63 


64 


54.86 


32.96 


54.71 


33.20 


54.57 


33.44 


54.42 


33.68 


64 


65 


55.72 


33.48 


55.57 


33.72 


55.42 


33.96 


55.27 


34.20 


66 


66 


56.57 


33.99 


56.42 


34.24 


56.27 


34.48 


56.12 


34.73 


66 


67 


57.43 


34.51 


57.28 


34.76 


57.13 


35.01 


56.98 


35.26 


67 


68 


58.29 


35.02 


58.13 


35.28 


57.98 


35.53 


57.82 


35.78 


68 


69 


59.14 


35.54 


58.99 


35.80 


58.83 


36.05 


58.67 


36.31 


69 


70 

71 


60.00 


36.05 


59.84 


36.31 


59.68 


36.57 


59.52 


36.83 


70 
71 


60.86 


36.57 


60.70 


36.83 


60.54 


37.10 


60.37 


37.36 


72 


61.72 


37.08 


61.55 


37.35 


61.39 


37.62 


61.23 


37.89 


72 


73 


62.57 


37.60 


62.41 


37.87 


62.24 


38.14 


62.08 


38.41 


73 


74 


63.43 


38.11 


63.26 


38.39 


63.10 


38.66 


62.93 


38.94 


74 


75 


64.29 


38.63 


64.12 


38.91 


63.95 


39.19 


63.78 


39.47 


75 


76 


65.14 


39.14 


64.97 


39.43 


64.80 


39.71 


64.63 


39.99 


76 


77 


66.00 


39.66 


65.83 


39.95 


65.65 


40.23 


65.48 


40.52 


77 


78 


66.86 


40.17 


66.68 


40.46 


66.51 


40.75 


66.33 


41.04 


78 


79 


67.72 


40.69 


67.54 


40.98 


67.36 


41.28 


67.18 


41.57 


79 


80 
81 


68.57 


41.20 


68.39 


41.50 


68.21 


41.80 


68.03 


42.10 


80 
81 


69.43 


41.72 


69.25 


42.02 


69.06 


42.32 


68.88 


42.62 


82 


70.29 


42.23 


70.10 


42.54 


69.92 


42.84 


69.73 


43.15 


82 


83 


71.14 


42.75 


70.96 


43.06 


70.77 


43.37 


70.58 


43.68 


83 


84 


72.00 


43.26 


71.81 


43.58 


71.62 


43.89 


71.43 


44.20 


84 


85 


72.86 


43.78 


72.67 


44.10 


72.47 


44.41 


72.28 


44.73 


85 


86 


73.72 


44.29 


73.52 


44.61 


73.33 


44.93 


73.13 


45.25 


86 


87 


74.57 


44.81 


74.38 


45.13 


74.18 


45.46 


73.98 


46.78 


87 


88 


75.43 


45.32 


75.23 


45.65 


75.03 


45.98 


74.83 


46.31 


88 


89 


76.29 


45.84 


76.09 


46.17 


75.88 


46.50 


75.68 


46.83 


89 


90 
91 


77.15 
78.00 


46.35 
46.87 


76.94 


46.69 


76.74 


47.02 


76.53 


47.36 


90 
91 


77.80 


47.21 


77.59 


47.55 


77.36 


47.89 


92 


78.86 


47.38 


78.65 


47.73 


78.44 


48.07 


78.23 


48.41 


92 


93 


79.72 


47.90 


79.51 


48.25 


79.30 


48.59 


79.08 


48.94 


93 


94 


80.57 


48.41 


80.36 


48.76 


80.15 


49.11 


79 93 


49.47 


94 


95 


81.43 


48.93 


81.22 


49.28 


81.00 


49.64 


80 78 


49. 9S 


95 


90 


82.29 


49.44 


82.07 


49.80 


81.85 50.16 


81.63 


50.52 


96 


97 


83.15 


49.96 


82.93 


30.32 


82.71 


50 .68 


82.48 5:. 04 


97 


98 


84.00 


50.47 


83.78 


50.94 


83.56 


51.20 


83.33 


51.57 


98 


99 


84.86 


50.99 


84.64 


51.36 


84.41 


51.73 


84.18 


52.10 


99 


100 

Q 


85.72 


51.50 


85.49 


51.88 


85.26 


52.25 


85.04 


62.62 


100 

g 

01 

°1 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


59 I 


)eg. 


581 Deg. 


58i Deg. 


58i Deg. 



136 



TRAVERSE TABLE, 



B 
S 


32 Deg. 


32i Deg. 


3^ Deg. 


321 Deg. 


S' 

p 

3 
r> 
a 

1 


Lat. 


Dep, 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


" 0.85 


0.63 


0.85 


0.53 


0.84 


0.54 


0.84 


0.64 


2 


1.70 


1.06 


1.69 


1,07 


1.69 


1,07 


1.68 


1.08 


2 


3 


2.54 


1.69 


2.54 


1.60 


2.53 


1.61 


2.52 


1.62 


3 


4 


3.39 


2.12 


3.38 


2.13 


3.37 


2.15 


3.36 


2.16 


4 


5 


4.24 


2.65 


4.23 


2.67 


4.22 


2.69 


4.21 


2.70 


6 


6 


5.09 


3.18 


6.07 


3.20 


5.06 


3.22 


5,05 


3.25 


6 


7 


6.94 


3.71 


6.92 


3.74 


6.90 


3.76 


5.89 


3.79 


7 


8 


6,78 


4.24 


6,77 


•4.27 


6.75 


4.30 


6.73 


4.33 


8 


9 


7.63 


4.77 


7,61 


4.80 


7.59 


4.84 


7.67 


4.87 


9 


10 


8.48 


6.30 


8.46 


6.34 


8.43 


5.37 


8.41 


5.41 


10 


11 


9.33 


6.83 


9.30 


5.87 


9.28 


5.91 


9.25 


6.95 


11 


12 


10.18 


6.36 


10.15 


6,40 


10.12 


6.45 


10,09 


6.49 


12 


13 


11.02 


6.89 


10.99 


6.94 


10.96 


6.98 


10.93 


7.03 


13 


14 


11.87 


7.42 


11.84 


7.47 


11.81 


7.52 


11.77 


7.57 


14 


15 


12.72 


7.96 


12.69 


8.00 


12.65 


8.06 


12.62 


8.11 


15 


16 


13.57 


8.48 


13.63 


8.54 


13.49 


8.60 


13.46 


8.66 


16 


17 


14.42 


9.01 


14.38 


9,07 


14.34 


9.13 


14.30 


9.20 


17 


18 


16.26 


9.64 


15,22 


9,61 


16,18 


9.67 


15.14 


9.74 


18 


19 


16.11 


10.07 


16.07 


10.14 


16.02 


10.21 


15,93 


10,28 


19 


20 


16.96 


10.60 


16.91 


10.67 


1.6.87 


10.75 


16,82 


10.82 


20 

21 


21 


17.81 


11.13 


17.76 


11.21 


17,71 


11.28 


17.66 


11.36 


22 


18. 6e 


11.66 


18.61 


11.74 


19.55 


11.82 


18.50 


11.90 


22 


23 


19.61 


12.19 


19.45 


12.27 


19.40 


12,36 


19.34 


12.44 


23 


24 


20.35 


12.72J 


20.30 


12.81 


20.24 


12.90 


20.18 


12.98 


24 


25 


21.20 


13.25 


21.14 


13.34 


21.08 


13.43 


21.03 


13.52 


25 


26 


22.05 


13.78 


21.99 


13.87 


21.93 


13.97 1 


21.87 


14,07 


26 


27 


22.90 


14.31 


22.83 


14.41 


22.77 


14.51 


22.71 


14.61 


27 


28 


23.75 


14.84 


23.68 


14.94 


23.61 


15.04 


23.55 


15.15 


28 


29 


24.59 


16.37 


24.. 53 


16.47 


24.46 


15.58 


24.. 39 


15.69 


29 


30 


25.44 


15.90 


26.37 


16.01 


25.30 


16.12 


25.23 


16.23 


30 


31 


26.29 


16.43 


26.22 


16.54 


26.15 


16.66 


26.07 


16,77 


31 


32 


27.14 


16.96 


27.06 


17.08 


26.99 


17.19 


26.91 


17.31 


32 


33 


27.99 


17.49 


27.91 


17.61 


27.83 


17.73 


27.75 


17.85 


33 


34 


28.83 


18.02 


28.75 


18.14 


28.63 


18.27 


2a. 60 


18.39 


34 


35 


29.68 


18.55 


29.60 


18.68 


29.52 


18.81 


29.44 


18.93 


35 


36 


30.53 


19.08 


30.45 


19.21 


30.36 


19.34 


30.28 


19.48 


36 


37 


31.38 


19.61 


31.29 


19.74 


31.21 


19.88 


31.12 


20,02 


37 


38 


32.23 


20.14 


32.14 


20.28 


32.05 


20.42 


31.96 


20.56 


38 


39 


33.07 


20.67 


32.98 


20.81 


32.89 


20.95 


32.80 


21.10 


39 


40 


33.92 


21.20 


33.83 


21.34 


33.74 


21.49 


33.64 


21.64 


40 


41 


34.77 


21.73 


34.67 


21.88 


34.58 


22.03 


34.48 


22.18 


41 


42 


35.62 


22.26 


35.52 


22.41 


35.42 


22.57 


35.32 


22.72 


42 


43 


36.47 


22.79 


36.37 


22.95 


36.27 


23.10 


36.16 


23.26 


43 


44 


37.31 


23.32 


37.21 


23.48 


37.11 


23.64 


37.01 


23.80 


44 


45 


38.16 


23.86 


38.06 


24.01 


37.96 


24.18 


37,85 


2^.34 


45 


46 


39.01 


24.38 


3S.90 


24.56 


38.80 


24.72 


38.69 


24.88 


46 


47 


39.86 


24.91 


39.75 


25.08 


39 ..64 


25.25 


39.53 


25.43 


47 


AS 


40.71 


25.44 


40.59 


25.61 


40.48 


26.79 


40.37 


25.97 


48 


49 


41.55 


26.97 


41.44 


26.15 


41.33 


26.33 


41.21 


26.51 


49 


50 


42.40 


26.50 


42.29 


26.68 


42.17 


26.86 


42.05 


27.05 


60 




Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat 




"i 












Q 


58 Deg. 


571 


Deg-. 


57^ Deg. 


57J Deg. 


Q 



TRAVESSE TABLE. 



137 



a 

5' 

«-► 
p 

? 
51 


32Deg. 


32i Deg. 


32i Deg. 


321 Deg. 


s 

51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


43.25 


27«03 


43.13 


27,21 


43,01 


27.40 


42.89 


27.59 


52 


44.10 


27.56 


43.98 


27.75 


43.86 


27.94 


43.73 


28.13 


62 


53 


44.95 


28.09 


44.82 


28,28 


44.70 


28.48 


44.58 


28,67 


63 


54 


45.79 


28.62 


45.67 


28.82 


45,54 


29.01 


45.42 


29.21 


64 


55 


46.64 


29.15 


46.51 


29.35 


46.39 


29.55 


46.26 


29.76 


56 


56 


47.49 


29.68 


47.36 


29.88 


47.23 


30.09 


47,10 


30.29 


66 


57 


48.34 


30.21 


48.21 


30.42 


48.07 


30.63 


47.94: 


30.84 


67 


68 


49.19 


30.74 


49.05 


30.95 


48.92 


31.16 


48.78 


31.38 


58 


59 


50.03 


31.27 


49.90 


31.48 


49.76 


31.70 


49.62 


31.92 


99 


60 
61 


50,88 


31.80 


50.74 


32.02 


50.60 


32.24 


50.40 


32.46 


60 
61 


51.73 


32.33 


51.59 


32.55 


51.45 


32.78 


51.30 


33.00 


62 


52.58 


32.85 


52.44 


33.08 


52.29 


33.31 


52.14 


33.54 


62 


63 


53.43 


33.38 


53,28 


33.62 


53.13 


33.85t 


62.99 


34,08 


63 


64 


54.28 


33.91 


54.13 


34.15 


53.98 


34.39 


53.83 


34.62 


64 


65 


55.12 


34.44 


54.97 


34.68 


54.82 


34.92 


54.67 


35.16 


65 


66 


55.97 


34.97 


55.82 


35.22 


55.66 


35.40 


55.51 


35.70 


66 


67 


56:82 


35.50 


56.66 


35.75 


56.51 


36.00 


56.35 


36,25 


67 


68 


57.67 


36.03 


57.51 


36.29 


57.35 


36.64 


67.19 


36.79 


68 


69 


58.52 


36.56 


58.36 


36.82 


58,19 


37.07 


,58.03 


37.33 


69 


70 
71 


59.36 


37.09 


59.20 


37,35 


59.04 


37.61 


58,87 


57.87 


70 
71 


60.21 


37,62 


60,05 


37.89 


59.88 


38.15 


59,71 


38.41 


72 


61.06 


38.15 


60.89 


38.42 


60.72 


38.69 


60,65 


38.95 


72 


73 


61.91 


38.68 


61.74 


38.95 


61.57 


59.22 


61.40 


39. 4S 


73 


n 


62.76 


39.21 


62.58 


39.49 


62.41 


39.76 


62.24 


40.03 


74 


75 


63.60 


39.74 


63.43 


40.02 


63.25 


40.30 


63.08 


40.57 


76 


76 


64.45 


40.27 


64.28 


40.55 


64.10 


40,63 


63.92 


41.U 


76 


77 


65.30 


40.80 


65.12 


41.09 


64.94 


41.37 


64.76 


41.65 


77 


78 


66.15 


41.33 


65.97 


41.62 


65.78 


41.91 


€5.00 


42.20 


78 


79 


67.00 


41.86 


66.81 


42.16- 


66.63 


42.45 


66.44 


42.74 


79 


80 
81 


67.84 


42.39 


67.66 


42.69 


67.47 


42.98 


67.28 


43.28 


80 
81 


68.69 


42.92 


68.50 


43.22 


68.31 


43.52 


68.12 


43.82 


82 


69.54 


43.45 


69.35 


43.76 


69.16 


44.06 


68.97 


44.36 


82 


83 


7D.39 


43.98 


70.20 


44.29 


70.00 


44.60 


69.81 


44.90 


83 


84 


71i24 


44.51 


71.04 


44.82 


70.84 


45.13 


70.65 


45.44 


84 


85 


72.08 


45.04 


71.89 


45.36 


71.69 


45.67 


71.49 


45.98 


85 


86 


72.93 


45.57 


72.73 


45.89 


72.53 


46.21 


72.33 


46.52 


86 


87 


73.78 


46. LO 


73.58 


46.42 


73.38 


46.75 


73.17 


47.06 


87 


88 


74.63 


45.63 


74.42 


46.96 


74.22 


47.28 


74.01 


47.61 


88 


89 


75.48 


47.16 


75.27 


47.49 


75.06 


47.82 


74.85 


48.15 


89 


:90 

91 


76.32 


47.69 


76.12 


48.03 


75.91 


48.36 
48.89 


75.69 


48.69 


90 
91 


77.17 


48.22 


76.96 


48.56 


76.75 


76.53 


49.23 


92 


78.02 


48.75 


77.81 


49.09 


77.59 


49.43 


77.38 


49 77 


92 


93 


78.81' 


49.28 


78.65 


49.63 


78.44 


49.97 


78.22 


50.31 


93 


94 


79,72 


49.81 


79.50 


50.16 


79.28 


50.51 


79.06 


50.85 


94 


95 


80.56 


50.34 


80. B4 


.50.69 


80.12 


51.04 


79.90 


51.39 


95 


96 


81.41 


50.87 


81.19 


51.23 


80.97 


51.58 


80.74 


51.93 


96 


97 


82.26 


51.40 


8^.04 


51.76 


81.81 


52.12 


81.58 


.52.47 


97 


98 


83.11 


51.93 


82.88 


52.29 


82.65 


52.66 


82.42 


53.02 


98 


99 


83.96 


.52.46 


83.73 


52.^3 


83.50 


63.19 


83.26 


53.56 


99 


100 

o 

.a 

Q 


84.80 


52.99 


84*57 


53.3& 


84.34 


53.73 


84.10 


54.10 


100 

© 
o 

c 

ei 


Dep. 


Lai. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


68 Deg. 


.671 Deg. 


57i Deg. 


57i Deg. 



138 



TRAVERSE TABLE. 



pi 

3 

a 

9 
~1 


33 Deg. 


33i Deg. 


33i Deg 


331 Deg. 


o 

X 

? 

■3 
1 


Lat. 


_ 1 
Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 
O.SC 


0.84 


0.54; 


0.84 


0.55 


0.83 


0.55 


b.83 


2 


1.68 


1.09 


1.67 


1.10 


1.67 


1.10 


1.66 


1.11 


2 


3 


2.52 


1.63 


2.51 


1.64 


2.. 50 


1.66 


2.49 


1 67 


3 


4 


3.35 


2.18' 


3.35 


2.19 


3.34 


2 21 


3.33 


2.22 


4 


5 4 19 


2 72 


4.18 


2.74 


4.17 


2.76 


4.16 


2.78 


5 


fl 5 03 3 27 


5.02 


3.29 


5.00 


3.31 


4.99 


3.33 


6 


7 . 5.87 3 81 ' 


5.85 


3.84 


5.84 


3.86 


5.82 


3.89 


7 


8: 6.71 


4.36 


6.69 


4.39 


, 6.67 


4.42 


6.65 


4.44 


8 


9 


7.55 


4.90 


7.53 


4.93 


7.50 


4.97 


7.48 


5.00 


f) 


10 
11 


8.39 
9.23 


5.45 


8.36 


5.48 
6.03 


8.34 
9.17 


5.52 
6.07 


8.31 


5.56 


10 
11 


5.99 


9.20 


9.15 


6.11 


12 


10.06 


6.54 


10.04 


6.58 


10.01 


6.62 


9.98; 6.67 


12 


13 


10.90 


7.08 


10.87 


7.13 


10.84 


7.18 :! 10.81 


7.22 


13 


14 


11.74 


7.62 


11.71 


7.68 


' 11.67 


7.73!: 11.64 


7.78 


14 


15 


12.58 


8.17; 


12.54 


8.22 


■ 12.51 


8.28 :j 12.47 


8.33 


15 


16 


13.42 


8.71 ' 


13.38 


8.77 


1 13.34 


8.83, 13.30 


8.89 


16 


17 


14.26 


9.26 


14.22 


9.32 


14.18 


9.38 ,: 14.13 


9.44 


17 


18 


15.10 


9.80 


15.05 


9.87 


, 15.01 


9.93': 14.97 


10.00 


18 


19 


15.93 


10.35 


15.89 


10.42 


15.84 


10.49 j 15.80 


10.56 


19 


20 


16.77 


10.89) 
11.44 


16.73 


10.97 
11.51 


1 16.68 


11.04 16.63 11.11 


20 
21 


21 


17.61 


17.56 


17.51 


11.59 17.46 11.67 


22 


18.45 


11.98 


18.40 


12.06 


1 18.35 


12.14 11 18.29 12.22 


22 


23 


19.29 


12.53 


19.23 


12.61 


! 19.18 


12.69 ' 19.12 12.78 


23 


24 


20.13 


13.07; 


20.07 


13.16 


20.01 


13.25 : 19.96 13.33 


24 


25 


20. P7 


13.62 


20.91 


13.71 


20.85 


13.80 20.79 13.89 


25 


26 


21.81 


14.16 


21.74 


14.26 


21.68 


14.35 1 21.62 ; 14.44 


26 


27 


22.64 


14.71 


22.58 


14.80 


22.51 


14.90 22.45 15.00 


27 


28 


23.48 


15.25 


23.42 


15.35 


23.35 


15.45 23.28 


15.56 


28 


29 


24.32 


15.79 


24.25 


15.90 


24.18 


16.01 ! 24.11 


16.11 


29 


30 


25.16 


16.34 


25 . 09 


16.45 


25.02 
25 . 85 


16.56 


24.94 


16.67 


30 
"31 


31 


26.00 


16.88 


25.92 


17.00 


17.11 


25.78 


17.22 


32 


26.84 


17.43 


26.76 


17.55 


26.68 


17.66 


26.61 


17.78 


32 


33 


27.68 


17.97 


27.60 


18.09 27.52 


18.21 


27.44 


18.33 


33 


34 


28.51 


18.52 


28.43 


18.64 28.35 


18.77 


28.27 


18.89 


34 


3.'i 


29.35 


19.06 


29.27 


19.19 29.19 


19.32 


29.10 ' 19.44 


35 


36 


30.19 


19.61 


30.11 


19.74 I 30.02 


19.87 


29.93 i 20.00 


36 


37 


31.03 


20.15 


30.94 


20.29 ! 30.85 


20.42 1 30.76 


20.56 


37 


38 


31.87 


20.70 


31.78 


20.84 


31.69 


20.97 : 31.60 


21.11 


38 


39 


32 71 


21.24 


32.62 


21.38 


32.52 


21.53 


32.43 


21.67 


39 


40 
41 


33.55 


21.79 


33.45 
34.29 


21.93 
22.48 


33.36 
34.19 


22.08 


33.26 


22.22 


40 
41 


34.39 


22.33 


22.63 


34.09 22.78 


42 


35.22 


22.87 


35.12 


23.03 


35.02 


23.18 : 34.92 i 23.33 


*2 


43 1 36.06 


23.42 


35.96 


23.58 


35. S6 


23.73 ; 35.75 


23.89 


43 


44 136.90 


23.96 


36.80 


24.12 


36.69 


24.29 , 36.58 


24.45 


44 


45 


37.74 


24.51 


37.63 


24.67 


37.52 


24.84 


37.42 


25 00 


45 


46 


38.58 


25.05 


38.47 


25.22 


38.36 


25.39 


38.25 25.56 


46 


47 


39.42 


25.60 


39.31 


25.77 


39.19 


25 . 94 


39.08 126.11 


47 


iS 


40.26 


26.14 


40.14 


26.32 


40.03 


26.49 


39.91 1 26.67 


48 


49 


41.09 


26.69 


40.98 


26.87 


40.86 


27.04 


40.74 27.22 


1 49 


50 


41.93 


27.23 


41.81 


27.41 


41.69 
Dep. 


27.60 


41.57 127.78 


! 50 

! 6 
i g 

OD 

5 


i 

s 
.a 


Dep. 


Lat. 


Dep. 


Lat 


Lat. 


Dep. Lat. 


1 

57 Deg. 


56| Deg. 


56i Deg. 


56J Deg. 



TRAVERSE TABLE. 



139 



o 
a 

a 
§ 

61 


33 Dog. 


33i Deg. 


33^ Deg. 


33i Deg. 


a 

5' 

<-► 

g 


CD 

61 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 

28.15 


Lat. 


Dep. 


42.77 


27.78 


42.65 


27.96 


42.53 


42.40 


28.33 


52 


43.61 


28.32 


43.49 


28.51 


43.36 


28.70 


43.24 


28.89 


52 


53 


44.45 


28.87 


44.32 


29.06 


44.20 


29.25 


44.07 


29.45 


53 


64 


45.29 


29.41 


46.16 


29.61 


45.03 


29.80 


44.90 


30.00 


64 


55 


46.13 


29.96 


46.00 


30.16 


45.86 


30.36 


46.73 


30.56 


56 


56 


46.97 


30.50 


46.83 


30.70 


46.70 


30.91 


46.56 


31.11 


66 


57 


47.80 


31.04 


47.07 


31.26 


47.63 


31.46 


47.39 


31.67 67 1 


68 


48.64 


31.59 


48.50 


31.80 


48.37 


32.01 


48.23 


32.22 6ft 1 


59 


49.48 


32.13 


49.34 


32.35 


49.20 


32.56 


49.06 


32 78 ( 


59 


60 
61 


50.32 


32.68 


50.18 


32.90 


50.03 

50.87 


33.12 1 
33.67 


49.89 33.33 


60 
61 


51.16 


33.22 


51.01 


33.45 


50.72 


33.89 


62 


52.00 


33.77 


61.85 


33.99 


61.70 


34.22 


51.55 


34.46 


62 


63 


52.84 34.31 


52.69 


34.54 


52.53 


34.77 


62.38 


35.00 


63 


64 


53.67 .34.86 


53.52 


35.09 


53.37 


36.32 


63.21 


35.56 


64 


65 


54.51 135.40 


54.36 


35.64 


54.20 


35.88 


54.05 


36.11 


65 


66 


55.351 35.95 


55.19 


36.19 


56.04 


36.43 


54.88 


36.67 


es 


67 


66.19 36.49 


56.03 


36.74 


56.87 


36.98 


55.71 


37.22 


67 


68 


57.03 


37.04 


66.87 


37.28 


56.70 


37.53 


56.64 


37.78 


68 


69 


57.87 


37.68 


57.70 


37.83 


57.54 


38.08 


57.37 38.33' 


69 


70 
71 


68.71 


38.12 


68.64 


38.38 


68.37 


38.64 


58.20 


38.89 
39.45 


70 

71 


59.55 


38.67 


69.38 


38.93 


59.21 


39.19 


59.03 


72 


60.38 


39.21 


60.21 


39.48 


60.04 


39.74 


59.87 


40.00 


72 


73 


61.22 


39.76 


61.05 


40.03 


60.87 


40.29 


60.70 


40.66 


73 


74 


62.06 


40.30 


61.89 


40.67 


61.71 


40.84 


61.53 


41.11 


74 


75 


62.90 


40.85 


62.72 


41.12 


62 . .54 


41.40 


62.. 36 


41.67 


76 


76 


63.74 


41.39 


63.56 


41.67 


63.38 


41.95 


63.19 


42.22 


76 


77 


64.58 


41.94 


64.39 


42.22 


64.21 


42.50 


64.02 


42.78 


77 


78 


65.42 


42.48 


65.23 


42.77 


66.04 


43.05 


64.85 


43.33 


78 


79 


66.25 


43.03 


66.07 


43.32 


65.88 


43.60 


65.69 


43.89 


79 


80 
81 


67.09 


43.57 


66.90 
67.74 


43.86 
44.41 


66.71 


44.16 


66.52 


44.45 


80 
81 


67.93 


44.12 


67.54 


44.71 


67.35 


45.00 


82 


68.77 


44.66 


68.58 


44.96 


68.38 


45.26 


68.18 


45.56 


82 


83 


69.61 


46.20 


69.41 


45.51 


69.21 


45.81 


69.01 


46.11 


83 


84 


70.46 


45 . 76 


70.25 


46.06 


70.05 


46.36 


69.84 


46.67 


84 


85 


71.29 


46.29 


71.08 


46.60 


70.88 


46.91 


70.67 


47.22 


86 


86 


72.13 


46.84 


71.92 


47.15 


71.71 


47.47 


71.51 


47.78 


86 


87 


72.96 


47.38 


72.76 


47.70 


72.55 


48.02 


72.34 


48.33 


87 


88 


73.80 


47.93 


73.69 


48.25 


73.38 ' 48.57 


73.17 


48.89 


88 


89 


74.64 


48.47 


74.43 


48.80 


74.22 49.12 


74.00 


49.45 


89 


90 
91 


75.48 


49.02 
49.56 


76.27 
76.10 


49.35 


75.05 

75.88 


49.67 
50.23 


74.83 


50.00 


90 
91 


76.32 


49.89 


75.66 


60.56 


93 


77.16 


.50.11 


76.94 


.50.44 


76.72 


50.78 


76.60 


51.11 92 J 


93 


78.00 


50.66 


77,77 


! 50.99 


77.55 


51.33 


77.33 


51.67 931 


94 78 83 i 51.20 


78.61 


61.. 54 


78.39 


51.88 


78.16 


52.22 


94 


95 79.67 151.74 


79.46 


52.09 


79.22 


52 43 


78.99 


62.78 


95 


96 


80.61 52.29 


80.28 


52.64 


80.05 


52.99 


79.82 


63.33 


96 


97 


81.36 52.83 


81.12 


.53.18 


80.89 


53.54 


180.65 


63.89 


97 


98 


82.19 i 53.37 


81.96 


53.73 


81.72 


54.09 


81.48 


64.45 981 


09 


83.03 1 63.92 


82.79 


54.28 


82.65 


54.64 


82.32 


65.00 99 1 


100 

i 


83.87 154.46 


83.63 


54.83 


83.39 


66.19 


83.16 
Dep. 


65.66 


_00 

a 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Lat. 


57 Deg. 


56^ Deg. 


56^ Deg. 


56i Dog. 





140 



TBAVERSE TABLS* 



5 


34Deg. 


Mi Deg. 


34^ 


Deg. 


34| Deg. 


O 

P 

3 

? 

1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0.83 


0.56 


0.83 


0.56 


0.82 


0.57 


0.82 


0.57 


2 


1.66 


1.12 


1.65 


1.13 


1.65 


1.13 


1.64 


1.14 


2 


8 


2.49 


1.68 


2.48 


1.69 


2.47 


1.70 


2.46 


1.71 


3 


4 


3.32 


2.24 


3.31 


2.25 


3.30 


2.27 


3.29 


2.28 


4 


5 


4.15 


2.80 


4.13 


2.81 


4.12 


2.83 


4.11 


2.85 


5 


6 


4.97 


3.36 


4.96 


3.38 


4.94 


3.40 


4.93 


3.42 


6 


7 


5.80 


3.91 


5.79 


3.94 


5.77 


3.96 


5.75 


3.99 


7 


8 


6.63 


4-47 


6.61 


4.50 


6.59 


4.-53 


6.57 


4.56 


8 


9 


7.46 


5.03 


7.44 


5.07 


7.42 


5.10 


7.39 


5.13 


9 


10 
11 


8.29 


5.59 


8.27 


5.63 


8.24 


5.66 


8.22 


5.70 


10 


9.12 


6.15 


9.09 


6.19 


9.07 


6.23 


9.04 


6.27 


11 


12 


9.95 


6.71 


9.92 


6.75 


9.89 


6.80 


9.86 


6.84 


12 


13 


10.78 


7.-27 


10.75 


7.32 


10.71 


7.36 


10.68 


7.41 


13 


14 


11.61 


7.83 


11.57 


7.88 


11.64 


7.93 


11.50 


7.93 


14 


15 


12.44 


8.39 


12.40 


8.44 


12.36 


8.50 


12.32 


8.55 


15 


16 


13.26 


8.95 


13.23 


9.00 


13.19 


9,06 


13,15 


9.12 


16 


17 


14.09 


9.51 


14.05 


9.57 


14.01 


9.63 


13.97 


9.69 


17 


18 


14.92 


10.07 


14.88 


10.13 


14.83 


10.20 


14.79 


10.26 


18 


19 


15.75 


10.62 


15.71 


10.69 


15.66 


10.76 


15.61 


10.83 


19 


20 
21 


16,58 


11.18 


16.53 


11.26 


16.48 


11.33 


16.43 


11.40 


20 


17.41 


11.74 


17.36 


11.82 


17.31 


11.89 


17.25 


11.97 


21 


22 


J8.24 


12.30 


18.18 


12.38 


18.13 


12.46 


18.08 


12.54 


22 


23 


19.07 


12.86 


19.01 


12.94 


18.95 


13.03 


18.90 


13. IL 


23 


24 


19.90 


13.42 


19.84 


13.51 


19.78 


13.59 


19.72 


13.68 


24 


25 


20.73 


13.98 


20.66 


14.07 


20.60 


14.16 


20.54 


14.25 


25 


26 


21.55 


14.54 


21.49 


14.63 


21.43 


14.73 


21.36 


14.82 


26 


27 


22.38 


15.10 


22.32 


15.20 


22.25 


15.29 


22.18 


15.39 


27 


28 


23.2] 


15 66 


23.14 


15.76 


23.08 


15.86 


23.01 


15.96 


28 


29 


24.04 


16.22 


23.97 


16.32 


23.90 


16.43 


23.83 


16.53 


29 


30 


24.87 


16.78 


24.80 


16.88 


24.72 


16.99 


24.63 


17.10 


30 
31 


31 


25.70 


17.33 


25.62 


17.45 


25.55 


17.56 


25.47 


17.67 


32 


26.53 


17.89 


26.45 


18.01 


26.37 


18.12 


26.29 


18.24 


32 


33 


27.36 


18.45 


27.28 


18.57 


27.20 


18.69 


27.11 


18.81 


33 


34 


28.19 


19.01 


28.10 


19.14 


28.02 


19.26 


27.94 


19.38 


34 


35 


29.02 


19.. 57 


28.93 


19.70 


28.84 


19.82 


28.76 


19.95 


35 


36 


29.85 


20.13 


29.76 


20.26 


29.67 


20.39 


29.58 


20.52 


36 


37 


30.67 


20.69 


30.58 


20.82 


30.49 


20.96 


30.40 


21.09 


37 


38 


31.50 


21.25 


31.41 


21.39 


31.32 


21.52 


31.22 


21.66 


38 


39 


32.33 


21.81 


32.24 


21.95 


32.14 


22.09 


32.04 


22.23 


39 


40 
41 


33.16 


22.. 37 


33.06 
33.89 


22.51 


.32.97 


22.06 


32.87 


^2.80 


40 


33.99 


22.93 


23.07 


33.79 


23.22 


33.69 


23.37 


41 


42 


34.82 


-23.49 


34.72 


23.64 


34.61 


23.79 


34.51 


23.94 


42 


43 


35.65 


24.05 


35.54 


24.20 


35.44 


24.36 


35.33 


24.51 


43 


44 


36.48 


24.60 


36.37 


24.76 


36.26 


24.92 


36.15 


25.08 


44 


45 


37.31 


25.16 


37.20 


25.33 


37.09 


25.49 


36.97 


25.65 


45 


46 


38.14 


25.72 


38.02 


25.89 


37.91 


26.05 


37.80 


20.22 


46 


47 


38.96 


26.28 


38.85 


26.45 


38.73 


26.62 


38.62 


26.79 


47 


48 


39.79 


26.84 


39.68 


27.01 


39.56 


27.19 


39.44 


27.36 


48 


49 


40.62 


27.40 


40.50 


^7.58 


40.38 


27.75 


40.26 


27.93 


49 


50 
g 


41.45 


27.96 


41.33 


28.14 


41.21 


28.32 


41.08 


28.50 


60 


Dep. 


Lat, 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


d 
o 

c 

5 


56 Deg^. 


55| 


Deg. 


55|1 


Deg. 


55i Deg. 



TRAVEKSE TABUS. 



141 



61 


34 Deg. 


34i Deg. 


34A Deg. 


341 Dig. 


D 
O 

a 
51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


42.28 


28.52 


42.16 


28.70 


42.03 


28.89 


41.90 


29.07 


52 


43.11 


29.08 


42.98 


29.27 


42.85 


29.45 


42.73 


29.64 


52 


53 


43.94 


29.64 


43.81 


29.83 


43.68 


30.02 


43.55 


30.21 


63 


54 


44.77 


30.20 


44.64 


30.39 


44.50 


30.59 


44.37 


30.78 


64 


55 


45.60 


30.76 


45.46 


30.95 


45.33 


31.15 


45.19 


31.35 


55 


56 


46.43 


31.31 


46.29 


31.52 


46.15 


31.72 


46.01 


31.92 


56 


57 


47.26 


31.87 


47.12 


32.08 


46.98 


32.29 


46.83 


32.49 


67 


58 


48.08 


32.43 


47.94 


32.64 


47.80 


32.85 


47.66 


33.06 


68 


59 


48.91 


32.99 


48.77 


33.21 


48.62 


33.42 


48.48 


33.63 


59 


60 
61 


49.74 


35.66 


49.60 


33.77 


49.45 


33.98 


49.30 


34.20 


60 
61 


50.57 


34.11 


60.42 


34.33 


50.27 


34.55 


.50.12 


34.77 


62 


51.40 


34.67 


51.26 


34.89 


51.10 


35.12 


50.94 


35.34 


62 


63 


62.23 


35.23 


52.08 


35.46 


51.92 


35.68 


51.76 


35.91 


63 


64 


53.00 


36.79 


52.90 


36.02 


52.74 


36.25 


62.59 


36.48 


64 


65 


53.89 


36.3.5 


53.73 


36,58 


63.67 


36.82 


53.41 


37.05 


65 


66 


54.72 


36.91 


54.55 


37.15 


64.39 


37,38 


54.23 


37.62 


66 


67 


55.55 


37.46 


.65.38 


a7.71 


65.22 


37,95 


55.05 


38.19 


67 


68 


50,37 


38.03 


56.21 


38.27 


56.04 


38.52 


55.87 


38.76 


68 


69 


57.20 


38.58 


67.03 


38.83 


56.86 


39.08 


56.69 


39.33 


69 


70 
71 


58.03 


39.14 


67.86 


39.40 


67.69 


39.66 


57.52 


39.90 


70 
71 


58.86 


39.70 


58.69 


39.96 


58.61 


40.21 


58.34 


40.47 


72 


59.69 


40.26 


69.51 


40.. 52 


59.34 


40.78 


59.16 


41.04 


72 


73 


60.52 


40.82 


60.34 


41.08 


60.16 


41.35 


59.98 


41.61 


73 


74 


61.35 


41.38 


61.17 


41.65 


60.99 


41.91 


60.80 


42.18 


74 


76 


62.18 


41.94 


61.99 


42.21 


61.81 


42.48 


61.62 


42.75 


75 


76 


63.01 


42.50 


62.82 


42.77 


62.63 


43.05 


62.45 


43.32 


76 


77 


63.84 


43.06 


63.66 


43.34 


63.46 


43.61 


63.27 


43.89 


77 


78 


64.66 


43.62 


64.47 


43.90 


64.28 


44.18 


64.09 


44.46 


78 


79 


65.49 


44.18 


05.30 


44.46 


65.11 


44.75 


64.91 


45.03 


79 


80 
81 


66.32 


44.74 


66.13 


45.02 


65.93 


45.31 


65.73 


45.60 


80 
81 


67.15 


45.29 


66.95 


46.59 


66.75 


45.88 


66.>55 


46.17 


82 


67,98 


'45.86 


67.78 


46.15 


67.58 


46.46 


67.37 


46.74 


82 


83 


68.81 


46.41 


68.61 


46.71 


68.40 


47.01 


68.20 


47.31 


83 


84 


69-. 64 


46.97 


69.43 


47.28 


69.23 


47.58 


69.02 


47.88 


84 


85 


70.47 


47.63 


70.26 


47.84 


70.05 


48.14 


69.84 


48.45 


85 


86 


71.30 


48.09 


71.09 


48.40 


70.87 


48.71 


70.66 


49.02 


86 


87 


72.13 


48.65 


71.91 


48.96 


71.70 


49.28 


71.48 


49.59 


87 


88 


72.96 


49.21 


72.74 


49.53 


72.52 


49.84 


72.30 


50.16 


88 


89 


73.78 


49.77 


73.57 


50,09 


73.36 


50.41 


73.13 


50.73 


89 


90 
91 


74.61 


50.33 


74.39 


60.65 


74.17 


60.98 


73.95 


51.30 


90 
91 


75.44 


50.89 


75.22 


51.22 


75.00 


61.54 


74.77 


51.87 


92 


76.27 


51.45 


76.05 


51.78 


75.82 


62.11 


75.69 


52.44 


92 


93 


77.10 


52.00 


76.87 


52.34 


76.64 


52.68 


76.41 


53.01 


93 


94 


77.93 


52.56 


77.70 


52.90 


77.47 


53.24 


77.23 


53.58 


94 


95 


78.76 


63.12 


78.53 


53.47 


78.29 


53.81 


78.06 


54.15 


95 


96 


79.69 


53.68 


79.36 


54.03 


79.12 


54.37 


78.88 


54.72 


96 


97 


80.42 


54.24 


80.18 


54.59 


79.94 


54.94 


79.70 


55.29 


97 


98 


81.25 


54.80 


81.01 


55.15 


80.76 


55.51 


80.52 


.55.86 


98 


99 


82.07 


55.36 


81.83 


55.72 


81.59 


56.07 


81. .34 


56.43 


99 


100 

i 
s 

CO 


82.90 


66.92 


82.66 


56.28 


82.41 


56.64 


82.16 


57.00 


100 

<u 

c 

ca 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


56 Deg. 


55| Deg. 


55^ Deg. 


55k Deg. 



142 



THAVEBSB TABLE. 





35 Deg. 


354 De^. 


35i Deg. 


25t 


Deg. 


5 
p 


Lai. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1 


0.82 


0.57 


0.82 


0.53 


0.81 


0.58 


0.81 


0.53 


1 


2 


1.64 


1.15 


1.63 


1.151 


1.63 


1.16 


1.62 


1.17 


2 


3 


2.46 


1.72 1 


2.45 


1.73 1 


2.44 


1.74 


2.43 


1.75 


3 


4 


3.29 


2.29 1 


3.27 


2.31 


3.26 


2.32 


3.25 


2.. 34 


4 


5 


4.10 


2.87, 


4.08 


2.89 


4.07 


2.90 


4.06 


2.92 


^ 


6 


4.91 


3.44 


4.90 


3.46 


4.83 


3.48 


4.87 


3.51 





7 


5.73 


4.01 I 


5.72 


4.04 1 


5.70 


4.06 


5.63 


4.09 


7 


8 


6.55 


4.59; 


6.53 


4.62 


6.51 


4.65 


6.49 


4.67 


8 


9 


7.37 


5.16 


7.35 


5.19 


7,33 


5.23 


7.30 


5.26 


9 


10 
11 


8.19 


5.74 1 


8.17 


5.77 


8.14 


5.81 


8.12 


5.34 


10 


9.01 


0.31 i 


8.93 


6.35; 


8.96 


6.39 1 


8.93 


6.43 


11 


12 


9.83 


6.88 


9.80 


6.93 


9.77 


6.971 


9.74 


7.01 


12 


13 


10.65 


7.46 


10.62 


7.50, 


10.53 


7.55 { 


IO..55 


7.60 


13 


14 


11.47 


8.03 


11.43 


8.03, 


11.40 


8.13 


11.36 


8.18 


14 


15 


12.29 


8.60 


12.25 


8.66 1 


12.21 


8.71 


12.17 


8.76 


15 


16 


13.11 


9.13 1 


13.07 


9.23 


13.03 


9.29 


12.99 


9.35 


16 


17 


13.93 


9.75 i 


13.83 


9.81 i 13.84 


9.87! 


13.80 


9.93 


17 


18 


14.74 


10.32 


14.70 


10.39 ii 14.65 


10.45 1 


14.61 


10.. 52 


18 


19 


15.56 


10.90 


15. .52 


10.97 1 15.47 
11.54!; 16.23 


11.03; 


15.42 


11.10 


19 


20 
21 


16.38 


11.47 


16.33 


11.61 'i 10.23 


11.63 


20 


17.20 


12.05 1 


17.15 


12.12 i; 17.10 


12.19:; 17.04 


12.27 


21 


22 


18.02 


12.62 


17.97 


12.70 <: 17.01 


12.78 ; 17.35 


12.85 


22 


23 


18.84 


13.19 


18.73 


13.27; 13.72 


13.36:1 13.67 


13.44 


23 


24 


19.66 


13.77; 


19.60 


13.85 i 19.54 


13.94 ;i 19.43 


14.02 


24 


25 


20.48 


14.34! 


20.42 


14.43 i 20.. 35 


14.52 20.29 


14.61 


25 


26 


21.30 


14.91 1 


21.23 


15.0' ; 21.17 


15.10 i 21.10 


15.19 


26 


27 


22.12 


15.49 i 


22.05 


15.58 21.03 


15.68 21.91 


15.77 


27 


28 


22.94 


16.06 


22.87 


16.16 22.80 


16.20, 22.72 


16.36 


23 


29 


23.76 


16.63, 


23.68 


16.74 23.61 


16.S4' -23.54 


16.94 


29 


30 


24.. 57 


17.21 1 


24.50 
25.32 


17.31 1 24.42 


17.42 24.35 


17.53 


39 


31 


25.39 


17.78; 


17.89 25.24 


18.00 25.16 


13.11 


31 


32 


26.21 


18.. 35 


26.13 


13.47; 26.05 


13.58 ; 25.97 


13.70 


32 


33 


27.03 


13.93 


26.95 


19.05 26.87 


19.16 ,i 26.78 


19.23 


33 


34 


27.85 


19.50: 


27.77 


19.62 27.68 


19.74 1:27.59 


19.36 


34 


35 


23.67 


20.03 ' 


23.53 


20.20 23.49 


20.32 1 23.41 


20.45 


35 


36 


29.49 


20.65 


29.40 


20.78 1 29.31 


20.91 |'29.22 


21.03 


36 


37 


30.31 


21.22' 


30.22 


21.35 ,30.12 


21.49 '30.03 


21,62 


37 


38 


31.13 


21.80 : 


31.03 


21.93 ' 30.94 


22.07 30.84 


22.20 


33 


39 


31.95 


22 . 37 j 


31.85 


22.51 31.75 


22.65 '31.65 


22.79 


39 


40 


32.77 


22.94! 
22.52 


.32.67 


23.09 


32.56 
33.33 


23.23 32.46 


23.37 
23.95 


40 
41 


41 


33.59 


33.48 


23.66: 


23.81 i 33.27 


42 


34.40 


24.09 


34.30 


24.24: 


.34.19 


24.39 ' .34.09 


24. S4 


42 


43 


35.22 


24.66' 


35.12 


24.82; 


35.01 


21:.97 1,34.90 


25.12 


43 


44 


86.04 


25.24 


,35.93 


25.39 1 


35.82 


25.55 !; 35.71 


25.71 


44 


45 


36.86 


25.81 


.36.75 


25.97. 


36.64 


26.13 , .36.52 


26.29 


45 


46 


37.63 


26.33 


, 37.57 


26.55! 


37.45 


26.71 1 37.33 


26.83 


46 


47 


33.50 


26.96 


' 33.33 127.13'! 33.26 


27.29 -33.14 


27.46 


47 


48 


39.32 


27.53 


39.20 


27.70 j .39.03 


27.87,; 33.96 


23.04 


48 


49 


40.14 


28 . 1 1 


40.02 


23.28 1 39.89 


23.45 !| 39.77 


123.63 


49 


60 


40.96 


28.63 


40.33 


23.86, 


40.71 
Dep. 


29.04 '40.53 


29.21 


50 


6 

1 

(5 


Dep. 


Lat. 


j Dep. 


Lat. 


Lat. I'r Dep. 


Lat. 


c 

5 


55 ] 


Deg. 


1 

54| Deg. 


54i 


Deg. '! 54t 

Ii 


Deg. 



TRAVERSE TABLE. 



143 



o 
a 

51 


35 Deg. 


35^ Deg. 


35i Deg. 


35 1 Deg. 


a 

to' 

r-» 
l» 
3 
o 
<x> 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep.. 


Lat, 


Dep. 


41.78 


29.25 


41.65 


29.43 


41.52 


29.62 


41.39 


29.80 


52 


42.60 


29.83 


42.47 


30.01 


42.33 


30.20 


42.20 


30.38 


52 


63 


43.42 


.30.40 


43.28 


30.59 


43.15 


30.78 


43.01 


30.97 


53 


54 


44.23 


30.97 


44.10 


31.17 


43.96 


31.86 


43.82 


31.55 


54 


55 


45.05 


31.55 


44.92 


31.74 


44.78 


31.94 


44.64 


32.13 


55 


56 


45.87 


32.12 


45.73 


32.32 


45.59 


32.52 


45.45 


32 . 72 


56 


57 


46.69 


32.69 


46.55 


32.90 


46.40 


33.10 


46.26 


33.. 30 


57 


58 


47.51 


.33.27 


47.37 


33.47 


47.22 


33.68 


47.07 


33.89 


58 


59 


48.33 


33.84 


48.18 


34.05 


48.03 


34.26 


47.88 


34.47 


59 


60 
61 


49.15 


34.41 


49.00 


34.63 


48.85 


34.84 


48.69 


35 . 05 


60 
61 


49.97 


34.99 


49.82 


35.21 


49.66 


35.42 


49.51 


35 . 64 


62 


50.79 


35 . 56 


60.63 


35.78 


50.48 


36.00 


50.32 


36.22 


62 


63 


51.61 


36.14 


51.45 


36.36 


51.29 


36.58 


61.13 


36.81 


63 


64 


52.43 


36.71 


52.27 


36.94 


.52.10 


37.16 


51.94 


37.39 


64 


65 


53.24 


37.28 


53.08 


37.51 


.52.92 


37.75 


52.75 


37.98 


65 


66 


54.06 


37.86 


53.90 


38.09 


53.73 


38.33 


53.50 


38 . 56 


60 


67 


54.88 


38.43 


54.71 


38.67 


54.55 


38.91 


54.38 


39.14 


67 


68 


55.70 


39.00 


55.53 


39.55 


55.36 


39.49 


"55.19 


39.73 


68. 


69 


56.. 52 


39.58 


56.35 


39.82 


56.17 


40.07 


56.00 


40.31 


69 


70 
71 


57.. 34 


40.15 


57. IG 


40 40 


56.99 


40.65 


.56.81 


40.90 


70 
71 


58.16 


40.72 


57.98 


40.98 


57.su 


41.23 


57.62 


41.48 


72 


58 . 98 


41.30 


58.80 


41.55 


58 . 65 


41.81 


58.43 


42.07 


72 


73 


59.80 


41.87 


59.61 


42.13 


59.43 


42.39 


.59.24 


42.65 


73 


74 


60.62 


42.44 


60.43 


42.71 


60.24 


42.97 


60.06 


43.23 


•74 


75 


61.44 


43.02 


61.25 


43.29 


61.06 


43.55 


60.87 


43.82 


75 


76 


62.26 


43.59 


62.06 


43.86 


61.87 


44.13 


61.68 


44.40 


76 


77 


63.07 


44.17 


62.88 


44.44 


62.69 


44.71 


62.49 


44.99 


77 


78 


63.89 


44.74 


63.70 


45.02 


63.50 


45.29 


63.30 


45.57 


78 


79 


64.71 


45.31 


64.51 


45 . 59 


04.32 


45 . 88 


64.11 


46.16 


79 


80 
81 


05.. 53 
66.35 


45.89 


65.33 


46.17 


65.13 


46.46 


64 . 93 
65.74 


46 . 74 


80 
81 


46.46 


66.15 


46 . 75 


65 . 94 


47.04 


47.32 


82 


67.17 


47.03 


66.90 


47.33 


66.76 


47.62 


66.55 


47.91 


82 


83 


67.99 


47.61 


67.78 


47.90 


67.57 


48.20 


67.36 


48.49 


83 


84 


08.81 


48 . 1 8 


68.60 


48.48 


68.39 


48.78 


68.17 


48.08 


84 


85 


C9.63 


48.75 


69.41 


49.06 


69.20 


49.36 


08.98 


49.66 


85 


86 


70.45 


49.33 


70.23 


49.63 


70.01 


49.94 


69.80 


50.25 


86 


87 


71.27 


49.90 


71.05 


50.21 


70.83 


50.. 52 


70.61 


60.83 


87 


88 


72.09 


50 47 


71 .86 


50.79 


71.64 


51.10 


71.42 


51.41 


88 


89 


72.90 


51.05 


72.68 


51.37 


72.46 


51.68 


72 . 23 


52.00 


89 


90 
91 


73 . 72 


51.62 


73 . 50 


51.94 


73.27 
74.08 


.52.26 
52.84 


73.04 


52.58 


90 
91 


74.54 


.52.20 


74.31 


52,. 52 


73.85 


53.17 


92 


75.36 


52.77 


75.13 


53.10 


74.90 


53.42 


74.66 


53.75 


92 


93 


70.18 


.53.34 


75.95 


53.67 


75.71 


54.01 


75.48 


64.34 


93 


94 


77.00 


.53.92 


76.76 


54.25 


76.. 53 


-54 . 59 


76.29 


54 . 92 


94 


95 


77.82 


.54.49 


77.58 


64.83 


77.34 


65.17 


77.10 


65.50 


95 


96 


78.64 


55.06 


78.40 


.55.41 


78.16 


55.75 


77.91 


56.09 


96 


97 


79.46 


55.84 


79.21 


55.98 


78 97 


56.33 


78.72 


56.67 


97 


98 


80.28 


.50.21 


80.03 


56.. 56 


70.78 


50.91 


79.53 


67.28 


98 


99 


81.10 


56.78 


80.85 


57.14 


80.60 


57.49 


80 .35 


57.84 


99 


J 00 

d 
u 
c 
a 

a 


81.92 


,57.36 


81.66 


57.71 


SI. 41 


58.07 


81.16 


58.42 


100 

t) 
c 
a 

*-> 

m 

s 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


L.it. 


55 Deg. 


54| Deg. 


54^ Deg. 


54i Deg.^ 



144 



TRAVERSE TABLE. 



9 

S' 

i 

1 


tiS Deg. 


m Deg. 


36^ 


Dog. 


36i Deg. 


5 

St 

§ 


Lat. 


Dep. 
6.69 


Lat. 


Dep. 


Lat. 


Dop. 


Lat. 


Dep. 


0.81 


0.81 


0.59 


0.80 


0.59 


0.80 


0.60 


2 


1.62 


1.18 


1.61 


1.18 


1.61 


1.19 


1.60 


1 20 


8 


3 


2.43 


1.76 


2.42 


1.77 


2.41 


1.78 


2.40 


1.79 


3 


4 


3.24 


2.35 


3.23 


2.37 


3.22 


2.38 


3.20 


2.39 


4 


5 


4.05 


2.94 


4.03 


2.96 


4.02 


2.97 


4.01 


2.99 


r» 


6 


4.85 3.53 


4.84 


3.55 


4.82 


3.57 


4.81 


3.59 


6 


7 


5.66 4.11 


5.65 


4.14 


5.63 


4.16 


5.611 4.19 


7 


8 


6 47 4.70 


6.45 


4.73 


6.43 


4.76 


6.41 


4.79 


8 


9 


7.28 5.29 


7.26 


5.32 


7.23 


5.35 


7.21 


5.38 


9 


10 
11 


8.09 5.88 
8.90 6.47 


8.06 


5 91 


8.04 


5.95 


8.01 
8.81 


5.98 


10 
11 


8.87 


6.50 


8.84 


6.54 


6.58 


13 


9.71 7.05 


9.68 


7.10 


9.65 


7.14 


9.61 


7.18 


12 


13 


10.52 


7.64 


10.48 


7.69 


10.45 


7.73 


10.42 


7.78 


13 


14 


11.33 


8.23 


11.29 


8.28 


11.25 


8.33 


11.22 


8.38 


14 


15 


12.14 


8.82 


12.10 


8.87 


12.06 


8.92 


12.02 


8.97 


15 


16 


12.94 


9.40 


12.90 


9.46 


12.86 


9.52 


12.82 


9.57 


16 


17 


13.75 


9.99 


13.71 


10.05 


13.67 


10.11 


13.62 


10.17 


17 


18 


14.56 


10.58 


14.52 


10.64 


14.47 


10.71 


14.42 


10.77 


18 


19 


15.37 


11.17 


15.32 


11.23 


15.27 


11.30 


15.22 


11.37 


19 


20 
21 


16.18 1 11.76 


16.13 


11.83 


16.08 


11.90 
12.49 


16.03 

16.83 


11.97 


20 
21 


16.99 


12.34 


16.94 


13.43 


16.88 


12.56 


22 


17. SO 


12.93 


17.74 


13.01 


17.68 


13.09 


17.63 


13.16 


22 


23 


18.61 


13.52 


18.55 


13.60 


18.49 


13.68 


18.43 


13.76 


23 


24 


19.42 


14.11 


19.35 


14.19 


19.29 


14.28 


19.23 


14.36 


24 


25 


20.23 


14.69 


20.16 


14.78 


20.10 


14.87 


20.03 


14.96 


25 


26 


21.03 


15.28 


20.97 


15.37 


20.90 


15.47 


20.83 


15.56 


26 


27 


21.84 


15.87 


21.77 


15.97 


21.70 


16.06 


21.63 


16.15 


27 


28 


22.65 


16.46 


22.58 


16.56 


23.51 


16.65 


33.44 


16.75 


28 


29 


23.46 


17.05 


23.39 


17.15 


23.31 


17.25 


33.34 


17.35 


29 


30 
31 


24.27 


17.63 


24.19 


17.74 


24.12 


17.84 


34.04 17.95 


30 
31 


25.08 


18.22 


25.00 


18.33 


24.92 


18.44 


34.84 18.55 


32 


25.89 


18.81 


25.81 


18.92 


25.72 


19.03 


35.64 19.15 


32 


33 


26.70 


19.40 


26.61 


19.51 


26.53 


19.63 


26.44 


19.74 


33 


34 


27.51 


19.98 


27.42 


20.10 


27.33 


20.22 


27.24 


20.34 


34 


35 


28.32 


20.57 


28.23 


30.70 


28.13 


20 . 82 


38.04 


20.94 


35 


36 


29.12 


21.16 


29.03 


31.29 


28.94 


21.41 


38.85 


21.54 


36 


37 


29.93 


21.75 


29.84 


21.88 


39.74 


23.01 


29.65 


22.14 


37 


38 


30.74 


22.34 


30.64 


22.47 


30.55 


32.60 


30.45 


22.74 


38 


39 


31.55 


22. P2 1 


31.45 


23.06 


31.35 


33.20 


31.35 


23.33 


39 


40 
41 


32.36 


23.51 


32.26 
33.06 


23.85 


33.15 
33.96 


33.79 
34.39 


33.05 


23.93 
24.53 


40 
41 


33.17 


24.10 


24.24 


32.85 


42 


33.98 


24.69 


33.87 


24.83 


33.76 


34.98 


33.65 


25.13 


42 


43 


34.79 


25.27 


34.68 


25.43 


34.57 


35.58 


34.45 


25.73 


43 


44 


35.60 


25.86 


35.48 


26.02 


35.37 


36.17 


35.36 


26.33 


44 


45 1 


36.41 36.45! 


36.29 


26.61 


36.17 


26.77 


36.06 


26.92 


45 


46 ! 


37.21 


27.04 


37.10 


27.20 


36.98 


27.36 


36.86 


27.52 


46 


47 


38.02 


27.63 


37.90 


27.79 


37.78 


27.96 


37.66 


28.12 


47 


48 


38.83 


28.21 


38.71 


28.38 


38.59 


38.55 


38.46 


28.72 


48 


49 


39.64 


28.80 


39.52 


28.97 


39.39 


29.15 


39.26 


29.32 


49 


50 


40.45 


29.39 
Lat. 


40.32 


29.57 


40.19 


29.74 


40.06 


29.92 


50 


Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 

















1 






Q , 54Deg. 


53} Deg. 


53i Deg. 


63i Deg. 


.S 1 



TRAVERSE TABLE. 



146 



o 

P 

3 
P 

51 


1 

36Deg. 


36i Dog. 


36^ Deg. 


361 Deg. 


P 

51 


Lat. j Dep, 


Lat. 


Dop. 


Lat 


Dep 


Lat. 


Dep. 

30.61 


141.26 29.98 


41.13 


30.16 


41.00 


30.34 


40 86 


52 


42 07 30.56 


41.94 


30.75 


41.80 


30.93 


41 67 


31.11 


52 


53 


42.88 


31.15 


42.74 


31.34 


42.60 


31.53 


42.47 


31.71 


53 


54 


43.69 


31.74 


43.55 


31.93 


43.41 


32.12 


43.27 


32.31 


54 


55 


44.50 


'32.33 


44.35 


32.52 


44.21 


32.72 


44.07 


32.91 


56 


56 


45.30 


32.92 


45.16 


33.11 


45.02 


33.31 


44.87 


33.51 


56 


57 


46.11 


33.50 


' 45.97 


33.70 


45.82 


33.90 


46.67 


34.10 


57 


58 


46.92 


34.09 


46.77 


34.30 


46.62 


34.60 


46.47 


34.70 


58 


69 


47.73 


34.68 


47.58 


34.89 


47.43 


35 09 


47.27 


35.30 


59 


GO 
61 


48.54 


35.27 


48.39 


35.48 


48.23 


35.69 


48.08 


35.90 


60 
61 


49.35 


35.85 


49.19 


36.07 


49.04 


36.28 


48.88 


36.60 


62 


50.16 


36.44 


50.00 


36.66 


49.84 


36.88 


49.68 


37.10 


62 


63 


50.97 


37.03 


50.81 


37.25 


60.64 


37.47 


50.48 


37.69 


63 


64 


51.78 


37.62 


51.61 


37.84 


51.45 


38.07 


51.28 


38.29 


64 


65 


52.59 


38,21 


52.42 


38.44 


52.25 


38.66 


62.08 


38.89 


65 


66 


53.40 


38.79 


53.23 


39.03 


53.05 


39.26 


62.88 


39.49 


66 


67 


54.20 


39.38 


54.03 


39.62 


53.86 


39.85 


63.68 


40.09 


67 


68 


65.01 


39.97 


.54.84 


40.21 


54.66 


40.45 


54.49 


40.69 


68 


69 


55.82 


40.56 


55.64 


40.80 


56.47 


41.04 


65.20 


41.28 


69 


70 

71 


56.63 


41.14 


66.45 


41.39 


56.27 


41.64 


56.09 


41.88 


70 

71 


57.44 


41.73 


57.26 


41.98 


67.07 


42.23 


66.89 


42.48 


72 


58.25 


42.32 


58.06 


42.57 


57.88 


42.83 


67.69 


43.08 


72 


73 


59.06 


42.91 


58.87 


43.17 


68.68 


43.42 


58.49 


43.68 


73 


74 


59.87 


43.50 


59.68 


43.76 


69.49 


44.02 


59.29 


44.28 


74 


75 


60.68 


44.08 


60.48 


44.35 


60.29 


44.61 


60.09 


44.87 


76 


76 


61.49 


44.67 


61.29 


44.94 


61.09 


46.21 


60.90 


45.47 


76 


77 


62.29 


45.26 


62.10 


45.53 


61.90 


45.80 


61.70 


46.07 


77 


78 


63.10 


45.85 


62.90 


46.12 


62.70 


46.40 


62.60 


46.67 


78 


79 


63.91 


46.43 


63.71 


46.71 


63., 50 


46.99 


63.30 


47.27 


79 


80 
81 


64.72 


47.02 


64.52 


47.30 


64.31 


47.59 


64.10 


47.87 


80 
81 


65.53 


47.61 


65.32 


47.90 


65.11 


48.18 


64.90 


48.46 


82 


66.34 


48.20 


66.13 


48.49 


65.92 


48.78 


66.70 


49.06 


82 


83 


67.15 


48.79 


66.93 


49.08 


66.72 


49.37 


66.60 


49.66 


83 


84 


67.96 


49.37 


67.74 


49.67 


67.. 52 


49.97 


67.31 


50.26 


84 


85 


68.77 


49.96 


68.55 


50.26 


68.33 


60.66 


68.11 


60.86 


85 


86 


69.58 


50.55 


60.35 


50.85 


69.13 


61.15 


68.91 


61.46 


86 


87 


70.38 


51.14 


70.16 


51.44 


69 94 


51.75 


69.71 


62.05 


87 


88 


71.19 


51.73 


70.97 


52.04 


70.74 


62.34 


70.51 


52.65 


88 


89 


72.00 


52.31 


71.77 


52.63 


71.64 


52.94 


71.31 


63.26 


89 


90 
91 


72.81 
73.62 


52.90 


72.58 


53.22 


72.35 


53.53 


72.11 


63.86 


90 
91 


53.49 


73.39 


53.81 


73.16 


54.13 


72.91 


64.45 


92 


74.43 


54.08 


74.19 


64.40 


73.95 


54.72 


73.72 


55.05 


92 


93 1 75.24 


.54.66 


75.00 


54.99 


74.76 


55.32 


74.52 


55.64 


93 


J 94 76.05 


55.25 


76.81 


65.. 58 


75.56 


55.91 


75.32 


56.24 


94 


96 , 76- 86 


55.84 


76.61 


66.17 


76.37 


56.51 


76.12 


56.84 


95 


96 ' 


77.67 


56.43 


77.42 


56.77 


77.17 


57.10 


76.92 


57.44 


96 


97 


78.47 


57.02 


78.23 


57.36 


77.97 


67.70 


77.72 


58.04 


97 


98 


79.28 


67.60 


79.03 


57.95 


78.78 


58.29 


78.62 


68.64 


98 


99 


80.09 58.19 


79.84 


58.54 


79.58 


58.89 


79.32 


59.23 


99 


100 

6 


80.90 58.78 


80.64 


59.13 


80.39 


59.48 


80.13 


59.83 


100 


Dep. 


L&t. 


Dep. 


Lat. 


Dep. 


Lat 


Dep 


Lat. 


1 
























54 Deff. 


531 Dog, 


53i Deg. 


53i Deg. 


_f. 



146 



TRAVERSE TADLK. 



O I 37 Deg. 



37i Deg. 



37j Deg. 



STi Deg, 



Lat. Dep. 



0.30 
1.60 
2 . 40 

3.:a 

3.99 
4.79 
5.59 
6.39 
7.19 
7.99 

8.73 
9. 53 
10. a3 
11.13 
11.93 
12.73 
13.53 
U.a3 
15.17 
15.97 

2r 16.77 

22 17.57 

23 13.37 

24 19.17 

25 19.97 

26 20.75 

27 21.56 
23 22.33 

29 23.16 

30 23.93 



1 
2 
3 
4 
5 
6 
7 
8 
9 

11 
12 
13 
U 
15 
16 
17 
18 
19 
20 



0.60 
1.20 
1.31 
2.41 
3.01 
3.61 
4.21 
4.31 
5.42 
6.02 
6.62 
7.22 
7.32 
3.43 
9.03 
9.63 
10.23 
10.33 
11-43 
12.04 

12.64 
13.24 
13.34 

14.44 
15.05 
15.65 
16.25 
16.35 
17.45 
13.05 



Lat. 


Dep. 


0.30 


0.61 


1.59 


1.21 


2.39 


1.32 


3.13 


2.42 


3.93 


3 . •j3 


4.73 


3 63 


O.Ol 


4.24 


6.37 


4.34 


7.16 


5 45 



Lat. Dep. Lat. Dep 



9.55 
10.35 
11.14 
11.94 
12.74 
13.53 
14.33 
15.12 
15.92 

16.72 
17.51 
13.31 
19.10 
19.90 



7.26 

7.37 

3.47 

9.03 

9.63 

10.29 

10.90 

1 1 . 50 

12.11 

12.71 

13.32 

13.92 

14.53 

15.13 



2') 


TO 


15 


74 


"^ _ 


_^ j 


-0 


34 


oo 


29 


:o 


95 


23 


03 


17 


00 


23 


33 


15 


16 



0.79 
1.59 
2 . 33 
3.17 
3.97 
4.76 
5.55 
6.-35 
7.14 
7.93 

3.73 
9.52 
10.31 
11.11 
11.90 
12.69 
13.49 
14.23 
15.07 
15.37 

16.66 
17.4.5 
13.25 
19.04 
19.33 
20.63 
21.42 
22.21 
23! 01 
23.30 



0.61 
1.22 
].33 
2 43 
3.04 
3.65 
4.26 
4.87 
5.43 
6.09 
6.70 
7.31 
7.91 
8.52 
9.13 
9.74 
10.35 
10.96 
11.57 
12.13 

12.73 
13.39 
14.00 
14.61 
15.22 
15.33 
16.44 
17.05 
17.65 
13.26 



0.79 
1. 58 
2.37 
3.16 
3.95 
4.74 
5.53 
6.33 
7.12 
7.91 



0.61 
1.22 
1.84 
2.45 
3.06 
3.57 
4.29 
4.90 
5.51 
6.12 



3.70 

9.49 
10.2-3 
11.07 
11.36 
12.65 
13.44 
14.23 
15.02 
15.31 



6.73 

7.35 

7.96 

3.57 

9.18 

9.30 

10.41 

1 1 . 02 

11.63 

12.24 



16.60 
17.40 
13.19 
13.93 
19.77 
20.56 
21.35 
22.14 
22.93 
23 . 72 



12.36 
13.47 
14.03 
14.69 
15.31 
15.92 
16.53 
17.14 
17.75 
13.37 



2 
3 

4 

6 
7 
3 
9 
10 

11 

12 
13 
14 
15 
16 
17 
13 
19 

11 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 




TRAVERSE TABI^. 



147 



o 


a- Deg. 


37i Deg. 


37i Deg. 


m i)^g^ p. 


on 










i 


3 
O 

a 

JvT 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 

31.06 


Lat. 


Dep. 


40.73 


30.69 


40.60 


30.87 


40.46 


40.33 31.22 


51 


52 


41.53 


31.29 


41.39 


31.48 


41.25 


31.66 


41.12 


31.84 


52 


53 


42.33 


31.90 


42.19 


32.08 


42.05 


.32.26 


41.91 


32.46 


53 


54 


43.13 


32.50 


42.98 


32.69 


42.84 


32.87 


42.70 


33.06 


54 


55 


43.92 


33.10 


43.78 


33.29 


43.63 


33.48 


43.49 


33.67 


56 


56 


44.72 


33.70 


44.58 


33.90 


44.43 


34.09 


44.28 


34.28 


66 


67 


45.52 


34.30 


45.37 


34.60 


45.22 


34.70 


45.07 


34.90 


57 


58 


46.32 


34.91 


46.17 


35.11 


46.01 


36.31 


45.86 


36.61 


58 


59 


47.12 


35.51 


46.96 


35.71 


46.81 


35.92 


46.66 


36.12 


59 


60 
'61 


47.92 


36.11 
36.71 


47.76 


36.32 


47.60 


36.53 


47.44 


36.73 


60 
61 


48.72 


48.66 


36.92 


48.39 


37.13 


48.23 


37.35 


62 


49.52 


37.31 


49.35 


37.53 


49.19 


37.74 


49.02 


37.96 


62 


63 


50.31 


37.91 


50.16 


38.13 


49.98 


38.36 


49.81 


38.57 


63 


64 


51.11 


38.52 


50.94 


38.74 


50.77 


38.96 


.50.60 


39.18 


64 


65 


51.91 


39.12 


61.74 


39.34 


61.57 


39.57 


61.39 


39.79 


66 


66 


52.71 


39.72 


52.54 


39.96 


52.36 


40.18 


62.19 


40.41 


66 


67 


53.51 


40.32 


53.33 


40.56 


53.15 


40.79 


52.98 


41.02 


67 


68 


54.31 


40.92 


64.13 


41.16 


63.95 


41.40 


53.77 


41.63 


68 


69 


55.11 


41.53 


54.92 


41.77 


54.74 


42.00 


64.56 


42.24 


69 


70 
'71 


65.90 
56.70 


42.13 
42.73 


55.72 


42.37 


65.53 
66.33 


42.61 
43.22 


65.35 


42.86 


70 

"71 


56.62 


42.98 


66.14 


43.47 


72 


57.50 


43.33 


67.31 


43.68 


57.12 


43.83 


66.93 


44.08 


72 


73 


58.30 


43.93 


58 . 1 1 


44.19 


57.91 


44.44 


57.72 


44.69 


73 


74 


59.10 


44.53 


58.90 


44.79 


58.71 


46.06 


68.61 


45.30 


74 


75 


59.90 


45.14 


59.70 


46.40 


69.60 


45.66 


59.30 


45.92 


75 


76 


00.70 


45 . 74 


60.60 


46.00 


60.29 


46.27 


60.09 


46.63 


76 


77 


61.49 


46.34 


61.29 


46.61 


61.09 


46.87 


60.88 


47.14 


77 


78 


62.29 


46.94 


62.09 


47.21 


61.88 


47.48 


61.67 


47.76 


78 


79 


63.09 


47.54 


62.88 


47.82 


62.67 


48.09 


62.46 


48.37 


79 


80 

81 


63.89 
64.69 


48.15 


63.68 


48.42 
49.03 


63.47 


48.70 


63.26 
64.05 


48.98 


80 
81 


48.75 


64.48 


64.26 


49.31 


49.69 


82 


65.49 


49.35 


65.27 


49.63 


66.05 


49.92 


64.84 


60.20 


82 


83 


66.29 


49.95 


66.07 


50.24 


65.86 


50.53 


66.63 


50.81 


83 


84 


67.09 


50.55 


66.86 


.50.84 


66.64 


61.14 


66.42 


51.43 


84 


85 


67.88 


51.15 


67.66 


51.45 


67.43 


51.74 


67.21 


52.04 86 1 


86 


68.68 51.76 


68.46 


62.06 


68.23 


62.36 


68.00 


52.65 861 


87 


69.48 52.36 


09.26 


.62.66 


69.02 


52.96 


68.79 


53.26 


87 


88 


70.28 


52.96 


70.05 


63.27 


69.82 


63.67 


69.58 


63.88 


88 


89 


71.08 


53.56 


70.84 


63.87 


70.61 


64.18 


70.37 


64.49 


89 


90 
91 


71.88 


54.16 


71.64 


64.48 


71.40 


54.79 


71.16 


55.10 


90 
91 


72.68 


54.77 


72.44 


56.08 


72.20 


65.40 


71.95 


56.71 


92 


73.47 


55.37 


73.23 


65.69 


72.99 


66.01 


72.74 


56.32 


92 


93 74.27 55.97 


74.03 


56.29 


73.78 


56.61 


73.63 


56.94 


93 


94 


75.07 


56.67 


74.82 


66.90 


74.68 


57.22 


74.32 


67.55 


94 


95 


75.87 


57.17 


75.62 


57.50 


75.37 


57.83 


75.12 


68.16 


95 


96 


76.67 


57.77 


76.42 


68.11 


76.16 


68.44 


76 91 


68.77 


96 


97 


r7.47 


58.38 


77.21 


58.7] 


76.96 


69.06 


76 70 


69.39 


97 


98 


78.27 


58.98 


78.01 


59.32 


77.75 


69.66 


77.49 


60.00 


98 


99 


79.06 


.59.68 


78.80 


59.92 


78.54 


60.27 


78.28 160.61 


99 


100 

i 

S 


79.86 


60.18 


79.60 
Dep. 


60.53 


79.34 


60.88 


79.07 


61.22 


100 

1 


Dep. 


Lat. 


Lat. 


Dep. 


Lat. 


Dop. 


Lat. 


53 I 


Deg. 


52J Deg. 


52i Deg. 


52i Deg. 



24 



148 



TRAVERSE TABLB. 



I 

1 


3fl Deg. 


m i^eg. 


38i Deg. 


381 Deg. 


1 Distance."" 


La4. 
"0:79" 


Dep. 

0.62 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.79 0.62 


0.78 


0.62 


0.78 


0.63 


2 1.58 


1.23 


1.57 1.24 


1.57 


1.24 


1.56 


1.25 


2 


3 2.36 


1.85 


2.36 1.86 


2.35 


1.87 


2.34 


1.88 


3 


4 3.151 


2.46 


3.14 2.48 


3.13 


2.49 


3.12 


2.50 


4 


5 3.94' 3.08 1 


3.93; 3.10 


3.91 


3 11 


3.90 


3.13 


6 


6 


4.73 3.69 


4.71 1 3.71 


4.70 


3.74 


4.68 


3.76 


3 


7 


5.52 4.31 


5.50 4.33 


5.48 


4.36 


5.46 


4.38 


7 


8 


6.30 4.93 


6.28 


4.95 


6.26 


4.98 


6.24 


5.01 


8 


9 


7.09 5 54 


7.07 


5.57 


7.04 


5.60 


7.02 


rj.63 


g 


10 
11 


7.88 


6.16 
6.77 


7.85 


6.19 


7.8£ 


6.23 


7.80 


b.iixt ID 


8.67 


8.64 


6.81 


8.61 


6.85 


8.58 


6.89 11 


12 1 9.46 


7.39 


9.42 


7.43 


9.39 


7.47 


9.36 


7.51 12 


13 


10.24 


8.00 


10.21 


8.05 


10.17 8.09 


10.14 


8.14 13 


14 


11.03 


8.62 


10.99 


8.67 


10.96 8.72 


10.92 


8.76 14 


.'5 


11.82 


9.23 


11.78 


9.29 


11.74 


9.34 


11.70 


9.39 


15 


16 


12.61 


9.85 


12.57 


9.91 


12.52 


9.96 


12.48 


10.01 


16 


17 


13.40 


10.47 


13.35 


10.52 


13.30 


10.58 


13.26 


10.64 


17 


18 


14.18 


11.08 


14.14 


11.14 


14.09 


11.21 


14.04 


11.27 


18 


19 


14.97 


11.70 


14.92 ; 11.76 


14.87 11.83 


14.82 


11.89 


19 


20 
21 


15.76 


12.31 


15.71 


12.38 


15.65 1 12.45 


15.60 


12.62 


20 


16.55 


12.93 


16.49 


13.00 


16.43 i 13.07 


16 38 


13.14 


21 


22 


17.34 


13.54 


17.28 


13.62 


17.22 


13.70 


17.16 


13.77 


22 


23 


18.12 14.16 


18.06 


14.24 


18.00 


14.32 


17.94 


14.40 


23 


24 


18.91 14.78 


18.85 


14.86 


18.78 


14.94 


18.72 


15.02 


24 


25 


19.70 


15.39 


19.63 


15.48 


19.57 


15.56 


19.50 


15.65 


25 


26 


20.49 


16.01 


20.42 


16.10 


20.35 


16.19 


20.28 


16.27 


26 


27 


21.28 16.62 


21.20 


16.72 


21.13 


16.8] 


21.06 


16.90 


27 


28 


22.06 17.24 


21.99 


17.33 


21.91 


17.43 


121.84 


17.53 


28 


29 


22.85 17.85 


22.77 


17.95 


22.70 


18.05 


22.62 


18.15 


29 


30 
31 


23.64 18.47 


23.56 


18.57 


23.48 


18.68 


23.40 


18.78 


30 
31 


24.43 


19.09 


24.34 


19.19 


24.26 


19.30 


24.18 


19.40 


32 


25.22 


19.70 


25.13 1 19.81 


25.04 


19.92 


24.96 


20.03 


32 


33 


26.00 


20.32 


25.92: 20.43 


25.83 


20.54 


25.74 


20.66 


33 


34 


26.79 


20.93 


26.70 21.05 


26.61 


21.17 


26.52 


21.28 


34 


35 


27.58 


21.55 


27.49 i 21.67 


27.39 


21.79 


27.30 


21.91 


35 


36 


28.37 


22.16 


28.27 i 22.29 


28.17 


22.41 


28.08 


22.53 


36 


37 


29.16 


22.78 


29.06 


22.91 


28.96 


23.03 


28.86 


23.16 


37 


38 


29.94 


23.40 


29.84 


23.53 


29.74 


23.66 


29.64 


23.79 


38 


39 


30.73 


24.01 


30.63 


24.14 


30.52 


24.28 


30.42 


24.41 


39 


40 


31.52 


24.63 


31.41 


24.76 


31.30 


24.90 


31.20 


25.04 


40 


41 


32.31 


25.24 


32.20 25.38 


,32.09 


25.52 


31.98 


25.66 


41 


42 


33.10 


25.86 


32.98 26.00 


32.87 


26.15 


32.76 


26.29 


42 


43 


33.88 26.47 


33.77 1 26.62 


33.65' 26.77 


33.53 


26.91 


43 


44 


34.67 


27.09 


34.55 27.24 


34.43 


27.39 


34.31 


27.54 44 


45 


35 46 


27.70 


35.34 27.86 


35.22 


38.01 


35.09 


28.17 46 


4€ 36.25 


28.32 


36.12 1 28.48 


36.00 


28.64 


35.87 


28.79 46 


47 37.04 


28.94 


36.91 


29.10 


36.78 


29 26 


36.65 


29.42 47 


48 ' 37 82 


29.55 


37.70 


29.72 


37.57 


29,88 


37 43 


30.04 , 48 


49 


38.61 .30.17 


38.48 


30.34 


38.35 


30.60 


38.21 


30.67 1 49 


60 

• 


39.40 


30.78 


39.27 


30.95 


39.13 


31 13 
Lat. 


38.99 
Dep. 


31.30 



i 

a 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 










' 


52 Deg. 


511 Deg. 


Slh Deg. 


Bl\ Dog. 


.9 



TRAVERSE TABLE. 



149 



a 
51 


38 Deg. 


381 Deg. 


38^ Deg. 


38| Deg. 


d 

P 
o 


Lat. 


Dep. 


I^at. 


D 


ep. 


Lat. 


Dep. 


Lat. 


D 


ep. 


40.19 


31 


.40 


40 


.05 


31 


.57 


39.91 


31 


.75 


39 


.77 


3] 


.92 


,51 


52 


40.98 


32 


.01 


40 


.84 


32 


.19 


40.70 


32 


.37 


40 


.55 


32 


.55 


62 


53 


41.76 


32 


.63 


41 


.62 


32 


.81 


41.48 


32 


.99 


41 


.33 


33 


.17 


53 


54 


42.55 


33 


.25 


42 


.41 


33 


.43 


42.26 


33 


.62 


42 


.11 


33 


.80 


64 


65 


43.34 


33 


.86 


43 


.19 


34 


.05 


43.04 


34 


.24 


42 


,89 


34 


.43 


55 


56 


44.13 


34,48 


43 


.98 


34 


.67 


43.83 


34 


.86 


43 


.67 


35 


.05 


56 


57 


44.92 


35 


.09 


44 


.76 


35 


.29 


44.61 


35 


.48 


44 


.45 


35 


.68 


57 


58 


45.70 


35 


.71 


45 


.55 


35 


.91 


45.39 


36 


.11 


45 


.23 


36 


.30 


58 


59 


46.49 


36 


.32 


46 


.33 


36 


.53 


46.17 


36 


.73 


46 


.01 


36 


.93 


59 


60 
61 


47.28 


36 


.94 


47 


.12 


37 


.15 


46.96 


37 


.35 


46 


.79 


37 


.56 


60 


48.07 


37 


.56 


47 


.90 


37 


.76 


47.74 


37 


.97 


47 


..57 


38 


.18 


61 


62 


48.86 


38 


.17 


48 


.69 


38 


.38 


48.52 


38 


.60 


48 


.35 


38 


.81 


62 


63 


49.64 


38 


.79 


49 


.47 


39 


.00 


49.30 


39 


.22 


49 


.13 


39 


.43 


63 


64 


50.43 


39 


.40 


50 


.26 


39 


.62 


50.09 


39 


.84 


49 


.91 


40 


.06 


64 


65 


51.22 


40 


.02 


51 


.05 


40 


.24 


50.87 


40 


.48 


50 


.69 


40 


.68 


65 


66 


52.01 


40 


.63 


51 


.83 


40 


86 


51.65 


41 


.09 


51 


.47 


41 


.31 


66 


67 


52.80 


41 


.25 


52 


.62 


41 


48 


52.43 


41 


.71 


52 


.25 


41 


.94 


67 


68 


53.68 


41 


.86 


53 


.40 


42 


10 


53.22 


42 


.33 


53 


.03 


42 


.56 


68 


69 


54.37 


42 


.48 


54 


.19 


42 


72 


54.00 


42 


95 


53 


.81 


43 


.19 


69 


70 
71 


55.16 


43 


.10 


54 


.97 


43 
43 


34 
96 


54.78 


43 


58 


54 


.59 


43 


.81 


70 
71 


55.95 


43 


.71 


55 


.76 


55.57 


44 


.20 


55 


.37 


44 


4 


72 


56.74 


44 


.33 


56 


.54 


44 


57 


56.35 


44 


82 


58 


.15 


45 


.07 


72 


73 


57.52 


44 


94 


57 


33 


45 


19 


57.13 


45 


44 


56 


.93 


45 


.69 


73 


74 


58.31 


45 


56 


58 


11 


45 


81 


57.91 


46 


07 


57 


71 


46 


.32 


74 


75 


59.10 


46 


17 


53 


90 


46 


43 


58.70 


46 


69 


58 


49 


46 


.94 


75 


76 


59.89 


46, 


79 


59 


68 


47 


05 


59.48 


47 


31 


59 


27 


47 


..57 


76 


77 


60.68 


47 


41 


60 


47 


47 


67 


60.26 


47 


93 


60 


05 


48 


20 


77 


78 


61.46 


48 


02 


61 


25 


48 


29 


61.04 


48 


56 


60 


.83 


48 


82 


78 


79 


62.25 


48 


64 


62 


04 


48 


91 


61.83 


49 


18 


61 


61 


49 


.45 


79 


80 
81 


63.04 


49 


25 


62 


83 


49 


53 


62.61 


49 


80 


62 


39 


50 


07 


80 
81 


63.83 


49 


87 


63 


61 


50 


15 


63.39 


50 


42 


63 


17 


50 


70 


82 


64.62 


50 


48 


64 


40 


50 


77 


64.17 


51. 


05 


63 


95 


51 


33 


82 


83 


65.40 


51 


10 


65 


18 


51 


38 


64.96 


51. 


67 


64 


73 


51 


95 


83 


84 


66.19 


51 


72 


65 


97 


52 


00 


65.74 


52. 


29 


65 


51 


52 


58 


84 


85 


66.98 


52 


33 


66 


75 


52 


62 


66.52 


52. 


91 


66 


29 


53 


20 


85 


86 


67.77 


52. 


95 


67. 


54 


53. 


.24 


67.30 


53. 


54 


67. 


07 


53. 


83 


86 


87 


68.56 


53. 


56 


68. 


32 


53. 


86 


68.09 


54. 


16 


67. 


85 


54. 


46 


87 


88 


69.34 


54. 


18 


63. 


11 


54. 


48 


68.87 


54. 


78 


68. 


63 


55. 


08 


88 


89 


70.13 


54. 


79 


G9. 


89 


55. 


10 


69.65 


55. 


40 


69. 


41 


55. 


71 


89 


90 
91 


70.92 


55. 


41 


70. 


68 


.55. 


72 


70.43 


56. 
.56. 


03 
65 


70, 


19 


56. 


33 


90 
91 


71.71 


56. 


03 


71. 


46 


56. 


34 


71.22 


70. 


97 


5G. 


96 


92 


72.50 


56. 


64 


72. 


25 


56. 


96 


72.00 


57. 


27 


71. 


75 


57. 


58 


92 


93 


73.28 


.57. 


26 


73. 


03 


57. 


58 


72.78 


57. 


89 


72. 


53 


.58. 


21 


93 


94 


74.07 


57. 


87 


73. 


82 


58. 


19 


73.57 


58. 


52 


73. 


31 


.58. 


84 


94 


95 


74.86 


58. 


49 


74. 


61 


58. 


81 


74.35 


59. 


14 


74. 


09 


59. 


46 


95 


96 


75.65 


59. 


10 


75. 


39 


59. 


43 


75.13 


59. 


76 


74. 


87 


60. 


09 


96 


97 


76.44 


59. 


72 


76. 


18 


60. 


05 


75.91 


60. 


33 


75. 


65 


60, 


71 


97 


98 


77.22 


60. 


33 


76. 


96 


60. 


67 


76.70 


61. 


01 


76. 


43 


61. 


34 


98 


99 


78.01 


60. 


95 


77. 


75 


61. 


29 


77.48 


61. 


63 


77. 


21 


61. 


97 


99 


100 

•-> 
« 

5 


78.80 


61. 


57 


78. 


53 


61. 


91 


78.26 


62. 


25 


77. 


99 


62. 


59 


100 

oJ 
u 

c 
«j 

«-» 

s 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 1 


Dep. 


Lat. 


52 Deg. 


511 Deg 


• 


5li Deg. 


5 


liDeg 





150 



TRAVEESE TAUL&. 



? 


39 Deg. 1' 39i Deg. 3^ Deg. 39j Deg. 


■ q 


1 


] 




Lftt ' Dep. Lat. 


: Dep. La-t. Dep. j La.t. ! Dep. 


s 

s 
"^1 


1 0.78 0.63 0.77 


: 0.63 0.77 0.64 


1 0.77] 64 


2' 1.55 1.26 1.55 


1.27 1.54 1.27 


1.54 1.28 


2 


3 2.33 1.89 2.32 


1.90 2.31 1,91 


2.311 1.92 


3 


4 3.11 2.52 3.10 


, 2.53 3.09 2.54 


3.08 2.56 


4 


5 3.89 3.15 3.87 


3.16 3.86 3.18 


3.84 3.20 


5 


6 4.66 3.78 4.65 


3. SO 4.63 3.82 


4.61 3.84 


6 


7 5.44 4.41 5.42 


4.43 5.40 4.45 \ 5.38 ! 4.48 


7 


8 6.22 5.03, 6.20 5.06 6.17 5.09 j 6.15 1 5.12 


8 


9 6.99 5.68 6.97, 5.6« 6.94 5.72 1 6.92 ^ 5.75 


9 


10 


7.77 6.29 7.74 6.33 7.72 6.36 


7.69 6.39 
8.46 7.03 


10 
11 


11 


8 . 55 6.92 8 . 52 


6.96 8.49 7.00 


12 9.33 7.55 9.29 


7.59 9.26 7.63 


9.23! 7.67 


12 


13 10.10 8.18 10.07 


1 8.23 10.03 8.27 


9.99 8.31 


13 


14 10.88 8.81 10.84 


1 8.86 10.80 i 8.91 


10.76 8.95 


14 


15 11.66 9.44 11.62 


9.49 11.57 9.54 


11.53 9.59 


15 


16 12.43 


10.07 12.39 


10.12 12.35 10.18 


12.30 10.23 


16 


17 13.21 


10.70 13.16 


10.76 13.12 i 10.81 1 13.07 10. a? 


17 


18 13.99 11.33 13.94 


11.39 13.89 i 11.45 13.84 11.51 


18 


19 14.77 11.96 14.71 


12.02 14 66 12.09 14.61 12.15 


19 


20 15.54 


12.59 15.49 


12.65 15.43 12.72:15.33 12.79 


20 

21 


21 16.32 


13.22 16.26 


13.29 16.20 13.36 16.15 13.43 


22 17.10 13.84 17.04 


13.92 16.93 13.99 16.91 14.07 


22 


23 17.87 14.47 17.81 


14.55 17.75 14.63 17.63 14.71 


23 


24 18.65 15.10,18.59 


15.18 18.52 15.27 18.45 15.35 


24 


25 19.43 15.73 19.36 


15.82 19.29 15.90 19.22 15.99 


25 


26 20.21 16.36 20.13 


16.45 20.06 16.54 19.99 16.63 


26 


27 20.98 16.99 20.91 


17.08 20.83 ; 17.17 ,20.76 17.26 
17.72 21.61 17.81 i 21.53 17.90 


27 


S3 21.76 17.62 21.68 


28 


29 22.54 18.25 22.46 


18.35 22.33 18.45 22.30 18.54 


29 


30 23.31 


18.88 23.23 


18.93 23.15 19.03 '23.07 19.18 


30 
31 


31 24.09 


19.51 24.01 


19.61 23.92 19.72 23.83 19.82 


32 24.87 20.14 24.78 


20.25 24.69 20.35 24.60 20.46 


32 


33 25.65 20.77 25.55 


20.88 25.46 120.99 25.37 21.10 


33 


34 26.42 21.40 26.33 


21.51 26.24:21.63 26.14 21.74 


M 


35 27.20 22.03 27.10 


22.14 27.01 22.26 26.91 22.33 


35 


36 27.98 22.66 27.83 


22.78 27.73 22.90 27.63 23.02 


36 


37 28.75 


23.23 28.65 


23.41 23.55 23.53 23.45 23.66 


37 


as 29.53 


23.91 29.43 


24.04 29.32 24.17 29.22 24.30 


33 


39 30.31 


24.54 30.20 


24.68 30.09 24.31 29.93 24.94 


39 


iO 31.09 
41 31.88 


25.17 30.93 


25.31 ■^■1 -« 25.44 ; 


30.75 25.53 ' 
31.. 52 26.22, 


to 

41 


25.50 31.75 


25.9-i 26.03 


42 32.64 


26.43 32.52 


26.57 .34. -x: 26.72 32.29 26.86 1 


y 


43 33.42 


27.06 33.. 30 


27.21 33.18 '27. 35 33.06 27.50 


43 


44 34.19 


27.69 .34.07 


27.84 .33.95 27.99 33.83 23.14! 


44 


45 .^i.97 


23.. 32 34. S5 


28.47 34.72 28.62 34.60 28.77 1 


45 


16 35.75 


28.95 35.62 


29.10 .35.49 29.26 35.37 29.41 


46 


47 36.53 


29.53 36.40 


29.74 36.27 29.90 36.14 30.05; 


47 


te 37.30 


30.21 37.17 


30.37 37.04 30.53 '36.90 30.69; 


43 


49 38.08 


30.!^ 37.95 


31.00 37.81 31.17 37.67 31.33 


49 


50 


33.86 


31.47 .38.72 


31.64, 33.53 31.80 ) 38.44 : 31 .97 i 


50 

d 

u 

■ 
3 


I' 


Dep. 


Lat Dep. | 


Lat. !' Dep. Lat. j! Dep. 


Lat. 


51 r 


( 
)eg. 50^ Deg. 5O5 De?. ! 

i 


50i Deg. 1 



TRAVEJISE TABLE. 



151 



f 


39 Dog. 


39i Dcg. 


39i Dog. 


391 Dog. 


s 


























61 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 


Dep, 


Lat. 


Dep. 


i 

51 


39.63 


32.10 


39.49 


32.27 


39.35 


32.44 


39.21 


32.61 


^2 


40.41 


32.72 


40.27 


32.90 


40.12 


33.08 


39.98 


33 25 


62 


03 


41.19 


33.35 


41.04 


33.53 


40.90 33.71 


40.75 


33 89 


53 


64 


41.97 


33.98 


41.82 


34.17 


41.67 134.35 


41.52 


34.53 


54 


66 


42.74 


34 61 


42.59 


34.80 


42.44 


34.98 


42.29 


35.1? 


55 


66 


43.62 36.24 


43.37 


35.43 


43.21 


35.62 


43.06 


35.81 


56 


67 


44.30 35.87 


44.14 


36.06 


43.98 


36.26 


43.82 


36.45 


57 


68 


45.07 


36.60 


44.91 


36.70 


44.75 


36.89 


44.69 


37.09 


58 


69 


46.85 


37. IS 


45.69 


37.33 


45.53 


37.53 


45.36 


37.73 


59 


60 
61 


46.63 


37.76 
38.39 


46.46 


37.96 


46.30 


38.16 


46.13 


38.37 


60 
61 


47 41 


47.24 


38.60 


47.07 


38 86 


46.90 


39.01 


62 


48.18 


39.02 


48.01 


39.23 


47.84 


39.44 


47.67 


39. 65' 


62 


63 


48.96 


39.65 


48.79 


39.86 


48.61 


40.07 


48.44 


40.28 


63 


64 


49.74 


40.28 


49.56 


40.49 


49.38 


40.71 


49.21 


40.92 


64 


66 


50.61 


40.91 


50.34 


41.13 


50.16 


41.35 


49.97 


41.56 


65 


66 


51.29 


41.54 


51.11 


41.76 


50.93 


41.98 


50.74 


42.20 


66 


67 


52.07 


42.16 


51.88 


42.39 


51.70 


42.62 


51.51 


42.84 


67 


68 


62.85 


42.79 


52.66 


43.02 


52.47 


43.25 


52.28 


43.48 


68 


69 


53.52 


43.42 


53.43 


43.66 


53.24 


43.89 


53.05 


44.12 


69 


70 
71 


54.40 


44.05 


54.21 


44.29 


54.01 


4.53 


53.82 


44.76 


70 
71 


55.18 


44.68 


54.98 


44.92 


54.79 


45.16 


54.59 


45.40 


72 


55.95 


45.31 


55.76 


45.55 


55.. 56 


45.80 


55.36 


46.04 


72 


73 


56.73 


45.94 


56.53 


46.19 


56.33 


46.43 


56.13 


46.68 


73 


74 


57.51 


46.57 


57.31 


46.82 


57.10 


47.07 


56.89 


47.32 


74 


75 


58.29 


47.20 


58.08 


47.45 


57.87 


47.71 


57.66 


47.96 


75 


76 


.59.06 


47.83 


58.85 


48.09 


58.64 


48.34 


58.43 


48.60 


76 


77 


59.84 


48.46 


59.63 


48.72 


59.42 


48.98 


59.20 


49.24 


77 


78 


60.62 


49.09 


60.40 


49.35 


60.19 


49.61 


59.97 


49.88 


78 


79 


61.39 


49.72 


61.18 


49.98 


60.96 


50.25 


60.74 


50.52 


79 


80 
81 


62.17 


50.35 


61.95 
62.73 


50.62 


61.73 
62.50 


60.89 


61.51 


51.16 


80 
81 


62.95 


50.97 


51.25 


51.52 


62.28 


51.79 


82 


63.73 


51.60 


63.50 


51.88 


63.27 


52.16 


63.04 


52.43 


82 


83 


64.50 


5'x{.23 


64.27 


52.51 


64.04 


52.79 


63.81 


53.07 


83 


84 


65.28 


52.86 


65.05 


53.15 


64.82 


53.43 


64.58 


53.71 


84 


85 


66.06 


53.49 


65.82 


53.78 


65.59 


54.07 


65.35 


54.35 


85 


86 


66.83 


54.12 


66.60 


54.41 


66.36 


54.70 


66.12 


54.99 


86 


87 


67.61 


54.75 


67.37 


55.05 


67.13 


55.34 


66.89 


55.63 


87 


88 


68.-^9 


55.38 


68.15 


55.68 


67.90 


55.97 


ff7.66 


56.27 


88 


89 


69.17 


56.01 


68.92 


56.32 


68.67 


56.61 


68.43 


56.91 


89 


90 
91 


69.94 


56.64 


69.70 


56.94 


69.45 
70.22 


57.25 


69.20 


57.55 


90 

91 


70.72 


57.27 


70.47 


57.58 


57.88 


69.96 


58.19 


92 


71.50 


57.90 


71.24 


58.21 


70.99 


58.52 


70.73 


58.83 


99 


93 


72.27 


58.53 


72.02 


58.84 


71.76 


59.16 


71.50 


59. 4~ 


93 


94 


73.05 


59.16 


72.79 


59.47 


72.53 


59.79 


72.27 


60.11 


94 


95 


73.83 


59.79 


73.57 


60.11 


73.30 


60.43 


73.04 


60.75 


95 


96 


74.61 


60.41 


74.34 


60.74 


74.08 


61.06 


73.81 


61.39 


96 


97 


75.38 


61.04 


75.12 


61.37 


74.85 


61.70 


74.58 


62.03 


97 


98 


76.16 


61.67 


75.89 


62.01 


75.62 


62.34 


75.35 


62.66 


98 


99 


76.94 


62.30 


76.66 


62.64 


76.39 


62.97 


76.12 


63.30 


99 


100 
« 


77.71 


62.93 


77.44 


63.27 


77.16 


63.61 


76.88 


63.94 


100 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


51 Deg. 


50| Deg. 


50i Deg. 


SOiDeg. 



153 



TRAVERSE TABIF. 



5 

s 

3 
o 
? 

1 


40 Deg. 


40^ Deg. 


40i Deg. 


401 Deg. 


5* 

r» 
P 
3 

p 


Lat. 


Dep, 

0.64 


Lat. 


Dep. ' 


Lat. 


Dep. 


Lat, 


Dep. 


0.77 


0.76 


0.65 


0.76 


0.65 


0.76 


0.65 


1 


2 


1.53 


1.29' 1.53 


1.29 


1.52 


1.30 


1.52 


1.31 


2 


3 


2.30 


1.93: 2.29 


1.94 


2.28 


1.95 


2.27 


1.96 


3 


4 


3.06 


2.57 :■ 3.05 


2.58 


3.04 


2.60 


3.03 


2.61 


4 


6 


3. S3 


3.21 


3.82 


3.23 


3.80 


3.25 


3.79 


3.26 


6 


6 


4.60 


3.80 i 


4.58 


3.88 


4,56 


3.90 


4.55 


3.92 


6 


7 


5.36 


4.50 


5.34 


4.52 


5.32 


4.55 


5.30 


4.57 


7 


8 


6.13 


5.14 


6.11 


5.17 


6.08 


5.20 


6.06 


5.22 


8 


9 


6.89 


5.79 i 


6. 87 


5.82 


0.84 


5.84 


6.82 


5.87 


9 


10 
11 


7.66 


6.43! 


7.63 
8.40 


6.46 j 


7.60 


6.49 


7.58 


6.53 


10 
11 


8.43 


7.07, 


7.11 


8.36 


7.14 1 


8.33 


7.18 


12 


9.19 


7.71 


9.16 


7.75 


9.12 


7.79 1 


9.09 


7.83 


12 


13 9.96 


8.36 9.92 


8.40 


9.89 


8.44! 


9.85 


8.49 


13 


14 


10.72 


9.00 1 10.69 


9.05 


10.65 


9.09 j 


10.61 


9.14 


14 


In 


11.49 


9.64: 11.45 


9.69 


11.41 


9.74 1 


11.36 


9.79 


15 


16 


12.26 


10.28 


12.21 


10,34 


12.17 


10.39 


12.12 


10.44 


16 


17 


13.02 


10.93 


12.97 


10.98 


12.93 


11.04 


12.88 


n.io 


17 


18 


13.79 


11.57 


13.74 


11.63 


13.69 


11.69 


13.64 


11.75 


18 


19 


14.55 


12.21 


14.50 


12.28 


14.45 


12.34 


14.39 


12.40 


19 


20 


15.32 


12.86 


15.26 


12.92 


15.21 


12.99 


15.15 
15.91 


13.06 


20 
21 


21 


16.09 


13.50 16.03] 


13.57 


15.97 


13.64 i 


13.71 


22 


16.85 


14.14 


16.79 


14.21 


16.73 


14.29; 


16.67 14.36! 


22 


23 


17.62 


14.78 


17.55 


14.86 


17.49 


14.94 1 


17.42 


15.01 


23 


24 


18.39 


15.43 


18.32 


15.51 


18.25 


15.59 1 


18.18 


15.67 


24 


25 


19.15 


16.07 


19.08 


16.15 


19.01 


16.24 


18.94 


16.32 


25 


26 


19.92 


16.71 


19.84 


16.80 


19.77 


16.89 


19.70 


16.97 


26 


27 


20.68 


17.36 


20.61 


17.45 


20.53 


17.54 


20.45 


17.62 


27 


28 


21.45 


18.00 


21.37 


18.09 


21,29 


18.18 


21.21 


18.28 


28 


29 


22.22 


18.64 


22.13 


18.74 


22.05 


18.83 


21.97 


18.93 


29 


30 
31 


22.98 


19.28 


22 . 90 


19.38 


22.81 


19.48 


22.73 


19. .58 


30 
31 


23.75 


19.93 


23.66 


20.03 


23.57 


20.13 


23.48 


20.24 


32 


24.51 


20.57 


24.42 


20.68 


24.33 


20.78 


24.24 


20.89 


32 


33 


25.28 


21.21 


25.19 


21.32 


25.09 


21.43 


25.00 


21.54 


33 


34 


26.05 


21.85 


25.95 


21.97 


25.85 


22.08 


25.76 


22.19 


34 


35 


26.81 


22.50 


26.71 


22.61 


26.61 


22.73 


26.51 


22.85 


•35 


30 


27.58 


23.14 


27.48 


23.26 


27.37 


23.38 


27.27 


23.50 


36 


37 


28.34 


23.78 


28.24 


23.91 


28.13 


24.03 


28.03 


24.15 


37 


38 


29.11 


24.43 


29.00 


24.55 


28.90 


24.68 


28.79 


24.80 


38 


39 


29.88 


25.07 


29.77 


25.20 


29.66 


25.33 


29.54 


25.46 


39 


40 
41 


30.64 


25.71 
26.35 


30.53 


25.84 


30.42 


25.98 


30.30 


26.11 


40 
41 


31.41 


31.29 


26.49 


31.18 


26. C3 


31.06 


26.76 


42 


32.17 


27.00 


32.06 


27.14 


31.94 


27.28 


31.82 


27.42 


42 


43 


32.94 


27.64 


32.82 


27.78 


32.70 


27.93 


32.58 


28.07 


43 


44 


33.71 


28.28 


33.58 


28.43 


33.46 


28.58 


33.33 


28.72 


44 


45 


34.47 


28.93 


34.35 


29.08 


34.22 


29.23 


34.09 


29.37 


45 


46 


35.24 


29.57 


35.11 


29.72 


34.98 


29.87 


• 34.85 


30.03 


46 


47 


36.00 


30.21 1 35.87 


30.37 


35.74 


30.52 


35.61 


30.68 


47 


48 


36.77 


30.85 


; 36.64 


31.01 


36.50 


31.17 


36.36 131.33 


48 


49 


37.54 


31.60 


1 37.40 


31.66 


37.26 


31.82 


37.12 


31.99 


49 


60 

g 


38.30 


32.14 


38.16 


32.31 


38.02 


32.47 


37.88 


32.64 


50 

e3 


c 

Q 


Dep. ( Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep, 


Lat. 


50 Deg. 


491 Deg. 


49i Deg. 


49i Deg. 



TKAV£BSE TABLC. 



153 



b 

03 
P 

3 
n 
o 

~6i 


40 Deg, 


40i Deg. 


40i Deg. 


40| Deg. 


Q 

to' 

r* 
P 

3 
n 
9 

51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


39.07 


32.78 


38.92 


32.95 


38.78 


33.12 


38.64 


33.29 


52 


39.83 


33.42 


39.69 


33.60 


39.. 54 


33.77 


39.39 


33.94 


52 


63 


40.60 


34.07 


40.45 


34.24 


40.30 


34.42 


40.15 


34.60 


53 


54 


4K37 


34.71 


41.21 


34.89 


41.06 


35.07 


40.91 


35.25 


54 


55 


42.13 


35.35 


41.98 


35.. 54 


41.82 


35.72 


41.67 


35.90 


55 


56 


42.90 


36.00 


42.74 


36.18 


42.. 58 


36.37 


42.42 


36.55 


56 


57 


43.66 


36.64 


43.50 


36.83 


43.34 


37.02 


43.18 


37.21 


57 


58 


44.43 


37.28 


44.27 


37.48 


44.10 


37.67 


43.94 


37.86 


68 


59 


45.20 


37.92 


45.03 


38.12 


44.86 


3S.32 


44.70 


38.51 


59 


60 
61 


45.96 


38.57 


45.79 


38.77 


45.62 


38.97 


45.45 


39.17 


60 


46.73 


39.21 


46.56 


39.41 


46.38 


39.62 


46.21 


39.82 


61 


62 


47.49 


39.85 


47.32 


40.06 


47.15 


40.27 


46.97 


40.47 


62 


63 


48.26 


40.50 


48.08 


40.71 


47.91 


40.92 


47.73 


41.12 


63 


64 


49.03 


41.14 


48.85 


41.35 


48.67 


41.56 


48.48 


41.78 


64 


65 


49.79 


41.78 


49.61 


42.00 


49.43 


42.21 


49.24 


42.43 


65 


66 


50.56 


42.42 


50.37 


42.64 


60.19 


42.86 


50. oa 


43.08 


66 


67 


51.32 


43.07 


51.14 


43.29 


50.95 


43.51 


50.76 


43.73 


67 


68 


52.09 


43.71 


51.90 


43.94 


51. 7r 


44.16 


51.51 


44.39 


68 


69 


52.86 


44.35 


52.66 


44.58 


52.47 


44.81 


52.27 


45.04 


69 


70 
71 


53.62 


45.00 


53.43 


45.23 


63.23 


45.46 


53.03 


45.69 


70 
71 


54.39 


45.64 


54.19 


45.87 


63.99 


46.11 


63.79 


46.35 


72 


55.16 


46.28 


54.95 


46.52 


54.75 


46.76 


54.. 54 


47.00 


72 


73 


55.92 


46.92 


55.72 


47.17 


55.51 


47.41 


55.30 


47.65 


73 


74 


56.69 


47.. 57 


56.48 


47.81 


56.27 


48.06 


56.06 


48.30 


74 


75 


57.45 


48.21 


57.24 


48.46 


57.03 


48.71 


56.82 


48.96 


75 


76 


58.22 


48.85 


58.01 


49.11 


57.79 


49.36 


.57.57 


49.61 


76 


77 


58.99 


49.49 


58.77 


49.75 


58.55 


50.01 


.58.33 


50.26 


77 


78 


59.75 


50.14 


59.53 


50.40 


59.31 


50.66 


59.09 


50.92 


78 


79 


60.52 


50.78 


60.30 


51.04 


60.07 


.51.31 


59 .85 


51.57 


79 


80 
81 


61.28 


51.42 


61.06 


51.69 


60.83 


51.96 


60.61 


52.22 


80 
81 


62.05 


52.07 


61.82 


52.34 


61.59 


52.61 


61.36 


52.87 


82 


62.82 


52.71 


62.59 


52.98 


62.35 


53.25 


62.12 


53.53 


82 


83 


63.58 


53.35 


63.35 


53.63 


63.11 


53.90 


62.88 


.54.18 


83 


84 


64.35 


53.99 


64.11 


54.27 


63.87 


54.55 


63.64 


54.83 


84 


85 


65.11 


54.64 


64.87 


54.92 


64 63 


55.20 


64.39 


55.48 


85 


86 


65.88 


55.28 


65.64 


55.57 


65 39 


55.85 


65.15 


56.14 


86 


87 


66.65 


55.92 


66.40 


56.21 


66 16 


56.50 


65.91 


56.79 


87 


88 


67.41 


56.57 


67.16 


56.86 


66 92 


57.15 


66.67 


.57-44 


88 


89 


68.18 


.57.21 


67.93 


57.50 


67 68 


57.80 


67.42 


58.10 


89 


90 
91 


68.94 


57.85 


68.69 


58.15 


68.44 


58.45 


68.18 


58.75 


90 


69.71 


58.49 


69.45 


58.80 


69.20 


59.10 


68.94 


59.40 


91 


92 


70.48 


59.14 


70.22 


59.44 


69.96 


59.75 


69.70 


60.05 


92 


93 


71.24 


59.78 


70.98 


60.09 


70.72 


60.40 


70.45 


60,71 


93 


94 


72.01 


60.42 


71.74 


60.74 


71.48 


61.05 


71.21 


61.36 


94 


95 


72.77 


61.06 


72.51 


6L.38 


72.24 


61.70 


71 .97 


62.01 


95 


96 


73.54 


61.71 


73.27 


62.03 


73.00 


62.35 


72.73 


62.66 


96 


97 


74.31 


6^.35 


74.03 


62.67 


73.76 


63.00 


73.48 


63.32 


97 


98 


75.07 


62.99 


74.80 


63.32 


74.52 


63.65 


74.24 


63.97 


98 


99 


75.84 


63.64 


75.56 


63.97 


75:28 


64.30 


75.00 


64.62 


99 


100 

i 

O 


76.fi0 


64.28 


76.32 


64.61 


76.04 


64,94 


75.76 


65.28 


100 


D^p. 


Lat. 


Dcp. 


tat. 


Dep. 


Lat. 


Dep. 


Lat. 


o 

c 

•-» 
m 


- 50 Deg. 


491 Deg. 


49i Deg. 


494 Deg. 



154 



TBAVEESi: TABLE. 



2" 





1 


41 Deg. 


4U Deg. '. 

i ' 


41 i Deg. 


41| Deg. 


a 

P 

3 

r. 
c 

1 


Lat, 


Dep. ; 


Lat. ! Dep. ' 


Lat. ! Dep, | 


Lat 


Dep. 


0.75 


0.66 


0.75 


0.66 


0.75 0.66 


0.75 


0.67 





1.51 


1.31 


1.50 


1.32 


1.50 1.33 


1.49 


1.33 


2 


3 


2.26 1.97 


2.26 


1.95 


2.25 1,99: 


2.24 


2.00 


3 


4 


3.02 1 2.62 


3.01 


2.64 


3.00 2,65 


2.98 


2.66 


4 


5 


3.77! 3.2s 


3.76 


3.30 


3.74; 3.31 


3,73 


3.33 


5 


6 


4.53, 3.94 


4.51 


3.96 


4.49 3.9S 


4.48 


4.00 


6 


7 


5.2s! 4.59 


5.26 


4.62 


5.24; 4.64 


5.22 


4.66 


7 


8 


6.04 5.25 


6.01 


5.27 


5 99; 5.30: 


5.97 


5.33 


8 


9 


6.79 5.90 


6.77 


5.93 


6.74 


5,?6; 


6.71 


5.99 


9 


10 
11 


7.55 1 6.53 


7.. 52 


6.59 


7.49 

8.24 


6.63 


7.46 


6.66 


10 
11 


S.30! 7.22 


8.27 


7,25 


7.29 


8.21 


7.-32 


12 


9.06 i 7.57 


9.02 


7.91 


8.99 


7.95: 


8.95 


7.99 


13 


13 


9. SI 1 8.53 


9.77 


8.57 


9.74 


8.61 1 


9.70 


8.66 


13 


14 


10.57; 9.15 


10.53 


9.23 


10.49 


9.28 1 


10.44 


9.32 


14 


15 


li.32: 9.S4 


11.28 


9.89 


11.23 


9,94 


11.19 


9.99 


15 


16 


12.0s 10.50 


12.03 


10.55 


11.95 i 10.601 


11.94 


10.65 


16 


IT 


12.53 11.15 


12.78 


11.21 


12.73 ; 11.26 1 


12. B8 


11.82 


17 


IS 


13. 5S ; 11. SI 


13.. 53 


11.87 


13.45 : 11.93: 


13.43 


n.99 


18 


19 


14.34 12.47 


14.2S 


12.53 


14.23 12.59! 


14.18 


12.65 


19 


20 ' 15.09 13.12, 


15.04 


13.19- 


14.95 


13.25; 

13.91 ; 


14.92 


13.:^ 


20 

21 


21 


15. S5 13.75 


15.79 13.55 


15.73 


15.67 


13.98 


22 ' 


16.60 14.43 


16.54 14.51 


16. 4S 14.53! 


16.41 


14.65 


22 


23 


17.35 15.C>9 


17.29 15.16 


17.23 ' 15.24- 


17.16 


15.32 


23 


24 


IS. 11 15.75 


IS. 04 15. S2 


17.97 1 15.90 


17.91 


15.93 


24 


25 


1S.S7 16.40 


IS. 80 16. 4S 


18.72 1 16.57 f 18.65 


16.65 


25 


26 


19.62 17.06 


19.35 17.14 


19.47; 17.23 1 19.40 


17.31 


26 


27- 


20.35 17.71 


20.30 17.50 


20.22 i 17.89!'; 20.14 


17.98 


27 


2S| 


21.13 IS. 37 


CI .05 l"^ .4-: 


20.97 ' 13. .55; 


20,89 


18.04 


28 


29 i 


21. S9 19.03 


.--.-. _ ."^ ■ - -- 


21 .72 19.22 ' 


21.6-4 


19.31 


29 


30 
31 


22. C4 19.65 
23.40 20.34 


- i . .; ' 1 0.7? 


22.47 ; 19.83 , 
23 . 22 ' 20 . 54 ; 


22.38 


19.98 


30 
31 


23.31 ; 20.44 


23.13 


20.6^1 


32 , 


24.15 20.99 


24.06 '21.10 


23.97 ' 21.20 1 


23.87 


21.31 


32 


33. 


24.91 21.65 


24.81 ; 21.76 


24.72 ; 21.87 


24.62 


21.97 


33 


34 


25.66 22.31 


25,56 ; 22.42 


25.46 22. .53 


25.37 


22.64 


34 


35 


26.41 22.96 


26.31 23.05 


26.21 23.19 


26.11 


23.31 


35 


36 


27.17 23.62 


27.07 23.74 


25.96 23.S5 1 


126.86 


23.97 


36 


37 


27.92 24.27 


27.82 2-1.40 


27.71 124.52! 


i27.ao 


24.64 


37 


38 


2-S.68 24.93 


28.57 25.05 


28.46 : 25.15 ; 


28.35 


25.. 30 


38 


39 


29.43 25.59 


29.32 25.71 


29.21 , 25.54: 


29.10 


25.97 


39 


40 
41 


30.19 26.24 
30.94 26.90 


30.07 i 26.37 
30.83 j 27.03 


29.96 I 25.r>6 , 
30 .'71 l27.17j 


i 29.84 


26.64 


40 

41 


30.. 59 


27.3i3 


42 


31.70 27.55 


31..5-8 127.69 


31.45 27. S3 


31.33 


27.97 


42 


43 


32.45 23.21 


32.33' 28.35 


32.21 2S.49 


32.08 


2-8.63 


43 


41 


33.21 2S.87 


33.08 29.01 


32.95 29.16 


132.83 


29.30 


44 


45 


33.96 29.52 


33.83 20. -:7 


33.70 29.82 


1 33.57 


29.97 


45 


46 


;U.72 30. IS 


34.58 ::.;; 


34.45 30.45 


•34,32 


30.63 


4^> 


47 


35.47 30. S3 


35.34 C;'.c<9 


35.20 31.14 


35.06 


31.30 


47 


48 


35. 2i^ 31.49 


36.09 ' 31.65 


35.95 31.81' 


35.81 


31.96 


48 


49 


36.93 32.15 


35.84 32.31 


36.70 ,32.47 


36.56 


32.63 


49 


50 


u 

c 


37.74. 


32.80 


.37.59 i 32.97 


37.45 ,33.13 
Dep. 1 Lat, 


37.30 


33.29 


50 



c 

/^ 


Dep. 


1 Lat. 


Dep. ; Lat. 


Dep. 


Lat. 


49 Deg, 


m 


Deg, 


, m Deg. 


48^ Deg. 

1 



TRAVERSE TABLE. 



165 



o 

p 
s 
o 
o 

51 


41 Deg. 


41i Deg. 


41^ Deg. 


41 1 Deg. 


m' 

3 
o 
p 

51 


Lat. 


Dep. 


Lat. 
38.34 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


38.49 


33.46 


33.63 


38.20 


33.79 


38.05 


33.96 


52 


39.24 


34.12 


39.10 


34.29 


38.95 


34.46 


.38.79 


34.63 


52 


53 


40.00 


34.77 


39.85 


34.95 


39.69 


35.12 


39.54 


35.29 


53 


54 


40.75 


35.43 


40.60 


35.60 


40.44 


35.78 


40.29 


35.96 


54 


55 


41.51 


36.08 


41.35 


36.26 


41.19 


36.44 


41.03 


36.62 


55 


56 


42.26 


36.74 


42.10 


36.92 


41.94 


37.11 


41.78 


.37.29 


56 


67 


43.02 


37.40 


42.85 


37.58 


42.69 


37.77 


42.53 


37.96 


57 


5S 


43 . 77 


33.05 


43.61 


38.24 


43.44 


38.45 


43.27 


33.62 


68 


59 


44.53 


38.71 


44.36 


38.90 


44.19 


39.09 


44.02 


39.29 


59 


(;o 

61 


45.28 


39.36 


45.11 
45.86 


39.56 


44.94 


39.76 


44.76 


39.95 


60 


46.04 


40.02' 


40.22 


45.69 


40.42 


45.51 


40.62 


61 


62 


4().79 


40.68 


46.61 


40.88 


46.44 


41.08 


46.26 


41.28 


62 


63 


47.55 


41.33 


47.37 


41.54 


47.18 


41.75 


47.00 


41.95 


63 


64 


48.30 


41.99 


48.12 


42.20 


47.93 


42.41 


47.75 


42.62 


64 


65 


49.06 


42.64 


48.87 


42.86 


48.68 


43.07 


48.49 


43.28 


65 


66 


49.81 


43.30 


49.62 


43.52 


49.43 


43.73 


49.24 


43.95 


66 


67 


50.57 


43.96 


50.37 


44.18 


50.18 


44.40 


49.99 


44.61 


67 


68 


51.32 


44.61 


51.13 


44.84 


50.93 


45.06 


50,73 


45.28 


68 


69 


52.07 


45.27 


51.88 


45.49 


51.68 


45.72 


51.48 


45.95 


69 


70 
71 


52.83 


45.92 


52,63 


46.15 


52.43 


46.38 


52.2,2 
.5^.97 


46.61 


70 
71 


53.58 


46.58 


53.38 


46.81 


53.18 


47.05 


47.28 


72 


54.34 


47.24 


54.13 


47.47 


53.92 


47.71 


53.72 


47.94 


72 


73 


55.09 


47.89 


54.88 


48.13 


54.67 


48.37 


54.46 


48.61 


73 


74 


55.85 


43.55 


55.64 


48.79 


55.42 


49.03 


55.21 


49.28 


74 


75 


56.60 


49.20 


56.39 


49.45 


56.17 


49.70 


.55.95 


49.94 


75 


76 


57.36 


49.86 


57.14 


50.11 


56.92 


50.. 36 


56.70 


50.61 


76 


77 


53.11 


50.52 


57.89 


50.77 


57.67 


51.02 


57.45 


51.27 


77 


78 


58.87 


51.17 


58.64 


51.43 


.58.42 


51.68 


58.19 


51.94 


78 


79 


59.62 


51.83 


59.40 


52.09 


59.17 


52.35 


58.94 


52.60 


79 


80 
81 


60.33 


52.48 


60.15 


52.75 


59.92 


53.01 


59.68 


53.27 


80 
81 


6 1 . r3 


53.14 


60.90 


53.41 


60.67 


53.67 


60.43 


53.94 


82 


61.89 


53.80 


61.65 


54.07 


61.41 


54.33 


61.18 


54.60 


82 


83 


62.64 


54.45 


62.40 


54.73 


62.16 


55.00 


61.92 


55.27 


83 


84 


63.40 


55.11 


63.15 


55.33 


62.91 


55.66 


62.67 


55.93 


84 


85 


64.15 


55.76 


63.91 


56.04 


63.66 


56.32 


63.41 


56.60 


85 


86 


64.90 


56.42 


64.66 


56.70 


64.41 


56.99 


64.16 


57.27 


86 


87 


65.66 


57.08 


65.41 


57.36 


65.16 


67.65 


64.91 


57.93 


R7 


88 


66.41 


57.73 


66.16 


58.02 


65.91 


58.31 


65.65 


58.60 


88 


89 


67.17 


58.39 


66.91 


58.68 


66.66 


58.97 


66.40 


59.26 


89 


90 
91 


67.92 


59.05 


67.67 


59.34 


67.41 


59.64 


67.15 


59.93 


90 


68.68 


59.70 


68.42 


60.00 


68.15 


60 .30 


67.89 


60.60 


91 


92 


69.43 


60.36 


69.17 


60.66 


68.90 


60.96 


68.64 


61.26 


92 


93 


70.19 


61.01 


69.92 


61.32 


69.65 


61.62 


69.38 


61.93 


93 


94 


70.94 


61.67 


70.67 


61.98 


70.40 


62.29 


70.13 


62.59 


94 


95 


71.70 


62.33 


71.43 


62.64 


71.15 


62.95 


70.88 


63.26 


95 


96 


72.45 


62.98 


72.18 


63.30 


71.90 


63.61 


71.62 


63.92 


96 


97 


73.21 


63.64 


72.93 


63.96 


72.65 


64.27 


72.37 


64.59 


97 


98 


73.96 


64.29 


73.68 


64.62 


73.40 


64.94 


73.11 


65,26 


98 


99 


74.72 


64.95 


74.43 


65.28 


74.15 


65.60 


73.86 


65.92 


99 


100 

o 

1 

.2 


75.47 


65. 6L 


75.18 


65.93 


74.90 


66.26 


74.61 


66.. 59 


100 

u 

,53 
O 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


49 Deg. 


48| Deg. 


^8^ DQg. 


48i Deg. 



156 



TBA VERSE TABLE. 



P 
3 
O 


42 Deg. 


424 Deg. 


42^ Deg, 


421 Deg. 


5 

re' 

P 

p 
1 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 1 

1 


Lat. 


Dep. 


1 


0.74 


0.67 


0.74 


0.67 


0.74 


0.68 


0.73 


0,63 


2 


1.49 


1.34 


1.48 


1.34 


1,47 


1.35 


1.47 


1.36 


2 


3 


2.23 


2.01 


2.22 


2.03 


2.21 


2.03 


2.20 


2.04 


3 


4 


2.97 


2.68 


2.96 


2.69 


2.95 


2.70 


2.94 


2.72 


4 


6 


3.72 


3.35 


3.70 


3.36 


3.69 


3.38; 


3.07 


3.39 


5 


6 


4.46 


4.01 


4.44 


4.03 


4.42 


4.05 1 


4.41 


4.07 


6 


7 


5.20 


4.68 


5.18 


4.71 


5.16 


4.73: 


5.14 


4.75 


7 


8 


5.95 


5.35 


5.92 


5.. 33 


5.90 


5.40 


5.87 


5.43 


8 


9 


6.69 


6.02 


6.66 


6.05 


6.64 


6.08 


6.61 


6.11 


9 


10 
11 


7.43 


6.69 


7.40 


6.72 


7,37 


6.76! 


7.34 


6.79 


10 


8.17 


7.36 j 


8.14 


7.40 


8.11 


7.43 1 


8.08 


7.47 


11 


12 


8.92 


8.03! 


8.88 


8.07 


8,85 


8.11 1 


8.81 


8.15 


12 


13 


9.66 


8.70 


9.62 


8.74 


9.58 


8.78, 


9.55 


8.82 


13 


14 


10.40 


9.37 


10.36 


9.41 


10.32 


9.46 


10.28 


9.. 50 


14 


15 


11.15 


10.04 


11.10 


10.09 


11.06 


10.13 


11.01 


10.18 


15 


16 


11.89 


10.71 


11.84 


10.76 


11.80 


10.81 


11.75 


10.86 


16 


17 


12.63 


11.38 


12.58 


11.43 


12.. 53 


11.48 


12.48 


11.-54 


17 


18 


13.33 


12.04 


13.32 


12.10 


13.27 


12.16 


13.22 


12.22 


IS 


19 


14.12 


12.71 


14.06 


12.77 1 


14.01 


12.84 


13.95 


12.90 


19 


20 
21 


14.86 


13.38 


14.80 


13.45 


14.75 


13.51 


14.69 


13.53 


20 


15.61 


14.05 


15.. 54 


14.12 


15.48 


14.19 


15.42 


14.25 


21 


22 


16.35 


14.72 


16.23 


14.79 


16.22 


14.86 16.16 


14.93 


22 


23 


17.09 


15.39 


17.02 


15.46 ; 


16,96 


15.54 k 16.89 


15.61 


23 


24 


17.84 


16.06 


17.77 


16.14 


17.69 


16.21 , 17.62 


16.29 


24 


25 


18.58 


16.73 


18.51 


16.81 


18.43 


16.89 


18.36 


16.97 


25 


26 


19.32 


17.40 


19.25 


17.48 


19.17 


17.57 


19.09 


17.65 


26 


27 


20.06 


18.07 


19.99 


18.15 


19.91 


18.24 


19.83 


18.33 


27 


2S 


20.81 


18.74 


20.73 


18.83 


20.64 


18.92 


20.. 56 


19.01 


28 


29 


21.55 


19.40 


21.47 


19.50; 


21.38 


19.59 


21.30 


19.69 


29 


30 


22.29 


20.07 


122.21 


20.17 j 


22-12 


20.27 


22.03 


20.36 


30 
31 


31 


23.04 


20.74 


122.95 


20.84' 


22.86 


20.94 


22.76 


21.04 


32 


23.78 


21.41 


23.69 


21.52 1 


23.59 


21.62 


23.50 


21.72 


32 


33 


24.. 52 


22.08 


24.43 


22.19 i 


24.-33 


22.29 


24.23 


22.40 


33 


34 


25.27 


22.75 


25.17 


22.86! 


25.07 


22.97 


24.97 


23.03 


34 


35 


26.01 


23.42 


25.91 


23.53 1 


25.80 


23.65 


25.70 


23.76 


.35 


36 


26.75 


24.09 


26.65 


24.21 


26.. 54 


24.. 32 


26.44 


24.44 


36 


37 


27 50 


24.76 


27.39 


24.88 


27-23 


25.00 


27.17 


25.12 


37 


38 


28.24 


25.43 


28.13 


25.55 


23.02 


25.67 


27.90 


25.79 


38 


39 


28.98 


26.10 


28.87 


26.22 


23.75 


26.35 


28.64 


26.47 


39 


40 
41 


29.73 


26.77 


29.61 


26.89 


29.49 


27.02 


29.. 37 


27,15 


40 
41 


30.47 


27.43 


i 30.35 


27.. 57 


30.23 


27.70 


30.11 


27.83 


42 


31.21 


28.10 


31.09 


28.24 


30.97 


28.37 


30.84- 


28.51 


42 


43 


31.96 


28.77 


31.83 


28. 9L 


31.70 


29.05 


31.58 


29.19 


43 


44 


32.70 


29.44 


32.57 


29, 5-3 


32.44 


29.73 


32-31 


29.87 


44 


45 


33.44 


30.11 


33.31 


30.26 


.33.18 


30.40 


33.04 


30.. 55 


45 


46 


34.18 


30.78 


34.05 


30.93 


33.91 


31.08 


33.78 


31.22 


46 


47 


34.93 


31.45 


34.79 


31.60 


34.65 


31.75 


34.51 


31.90 


47 


48 


35.67 


32.12 


35.63 


32.27 


35.39 


.32.43 


35.25 


32.53 


48 


49 


36.41 


32.79 


136.27 


32.95 


36.13 


33.10 


35^98 


.33.26 


49 


SO 


37.16 


33.46 


,37.01 


33.62 


36.85 


33.78 


.36.72 


.33.94 


50 

d 


8 

c 


Dep. 


Lat 


Dep. 


Lat. 


Dep. 


Lat, 


Dep. 


Lat 


48 Deg. 


47i Deg. 


47h Deg, 


41\ Deg. 



TBAVER8E TABUS 



157 



b 

w 

o 
? 

"61 


42 Deg. 


42| Deg, 


42i Deg, 


421 Deg. 


1* 

a 
a 
n 

61 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


37.90 


34.13 


37.75 


34.29 


37.60 


34.46 


37.45 


34.62 


52 


38.64 


34.79 


38.49 


34.96 


38.34 


35.13 


38.18 


35.30 


52 


53 


39.39 


35.46 


39.23 


35.64 


39.08 


35.81 


38.92 


35.98 


53 


64 


40.13 


36.13 


39.97 


36.31 


39.81 


36.48 


39.65 


36.66 


64 


55 


40.87 


36.80 


40.71 


36.98 


40.56 


37.16 


40.39 


37.33 


55 


56 


41.62 


37.47 


41.45 


37.65 


41.29 


37.83 


41.12 


38.01 


56 


57 


42.36 


38.14 


42.19 


38.32 


42.02 


38.51 


41.86 


38.69 


57 


68 


43.10 


33.81 


42.93 


39.60 


42.76 


39.18 


42.59 


39.37 


58 


69 


43.85 


39.48 


43.67 


39.67 


43.50 


39.86 


43.32 


40.05 


59 


60 
61 


44.59 


40.15 


44.41 


40.34 


44.24 


40.54 


44.06 


40.73 


60 
61 


45.33 


40.82 


45.15 


41.01 


44.97 


41 .21 


44.79 


41.41 


62 


46.07 


41.49 


45.89 


41.69 


45.71 


41.89 


45,53 


42.09 


62 


63 


46.82 


42.16 


46.63 


42,36 


46.45 


42.56 


46.26 


42.76 


63 


64 


47.58 


42.82 


47.37 


43,03 


47.19 


43.24 


47.00 


43.44 


64 


65 


48.30 


43.49 


48.11 


43.70 


47.92 


43.91 


47.73 


44.12 


65 


66 


49.05 


44.16 


48.85 


44.38 


48.66 


44.59 


48.47 


44.80 


66 


67 


49.79 


44.83 


49.59 


45.05 


49.40 


45.26 


49.20 


45.48 


67 


68 


50.53 


45.50 


50.33 


45.72 


50.13 


45.94 


49.93 


46.16 


68 


69 


51.28 


46.17 


51.07 


46.39 


.50.87 


46.62 


50.67 


46.84 


69 


70 
71 


52.02 


46.84 


51.82 


47.07 


51.61 


47.29 


51.40 


47.52 


70 
71 


52.76 


47.51 


52.56 


47.74 


52.35 


47.97 


52.14 


48.19 


72- 


53.51 


48.18 


53.30 


48.41 


53.08 


48.64 


6U.87 


48.87 


72 


73 


54.25 


48.85 


54.04 


49,08 


53.82 


49.32 


53.61 


49.. 55 


73 


74 


54.99 


49.52 


54.78 


49.76 


54.66 


49.99 


54.34 


50.23 


74 


75 


55.74 


50.18 


55.52 


50.43 


55.30 


50.67 


65.07 


50.91 


75 


76 


56.48 


.50.85 


56.26 


51.10 


56.03 


61.34 


55.81 


51.59 


76 


77 


67.22 


51.52 


57.00 


6.1.77 


56.77 


62.02 


56.. 54 


52.27 


77 


78 


67.97 


52.19 


57.74 


52.44 


57.61 


52.70 


57.28 


52.95 


78 


79 


58.7.1 


52.86 


58.48 


53.12 


58.24 


53.37 


58.01 


53.63 


79 


80 
81 


59.45 


53.53 


59.22 
59.96 


53.79 


58.98 


54.05 


58.75 


54.30 


80. 
81 


GO. 19 


54.20 


54.46 


59.72 


54.72 


59*48 


54.98 


82 


60.94 


54.87 


60.70 


55.13 


60.46 


55.40 


60.21 


55.66 


82 


83 


61.68 


55.54 


61.44 


55.81 


61.19 


56.07 


60.95 


56.34 


83 


84 


62.42 


56.21 


62.18 


56.48 


61.93 


66.75 


61.68 


57.02 


84 


85 


63.17 


56.88 


62.92 


.57.15 


62.67 


57.43 


62.42 


57.70 


85 


86 


63.91 


57.55 


63.66 


57.82 


63.41 


68.10 


63.15 


58.38 


86 


87 


64.65 


58.21 


64.40 


58.50 


64.14 


58.78 


63.89 


59.06 


87 


88 


65.40 


58.88 


65.14 


59.17 


64.88 


59.45 


64.62 


59.73 


88 


89 


66.14 


59.55 


65.88 


59.84 


65.62 


60.13 


65.35 


60.41 


89 


90 
91 


66.88 


60.22 


66.62 


60.51 


66.35 


60.80 


66.09 


61.09 


90 
91 


67.63 


60.89 


67.36 


61.19 


67.09 


61.48 


66.82 


61.77 


92 


68.37 


61.56 


68.10 


61.86 


67.83 


62.15 


67.56 


62.45 


92 


93 


69.11 


62.23 


68.84 


62.53 


68.57 


62.83 


68.29 


63.13 


93 


94 


69.86 


62.90 


69.58 


63.20 


69.30 


63.51 


69.03 


63.81 


94 


95 


70.60 


63.57 


70.32 


63.87 


70.04 


64.18 


69.76 


64.49 


95 


96 


71.34 


64.24 


71.06 


64.55 


70.78 


64.86 


70.49 


65.16 


96 


97 


72.08 


64.91 


71.80 


65.22 


71.52 


65.53 


71.23 


65.84 


97 


98 


72.83 


65.57 


72.54 


65.89 


72.25 


66.21 


71.96 


66.52 


98 


99 


73.57 


66.24 


73.28 


66.56 


72.99 


66.88 


72.70 


67.20 


99 


100 

6 
o 

a 


74.31 


66.91 


74.02 


67.24 


73.73 


67.56 


73»43 


67.88 


100 

8 

1 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


>48 Degi 


47} Deg. 


47i Deg. 


47i Deg, 



158 



TRAVESSE TAEIT:. 



5' 

P 
3 

o 

1 


! 
43Deg. 


43i Deg. 


43i Deg. 


431 Deg. 


D 

« 
? 

1 


Lat. 


Dep. 


Lat. 

1 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.73 


0.63 


0.73 


0.69 


i 0.73 


0.69 


0.72 


0.69 


2 


1.46 


1.36i 


1.46 


1.37 


! 1.45 


1.33 


1.44 


1.38 


2 


3 


2.19 


2.05^ 


2.19 


2.06 


1 2.13 


2.07 


i 2.17 


2.07 


3 


4 


2.93 


2.73 


2.91 


2.74 


2.90 


2.75 


j 2.39 


2.77 


4 


5 


3.66 


3.41 


3.64 
1 4.37 


3.43 


3.63 


3.44 


3.61 


3.46 


5 


6 


4.39 


4.09 


4.11 


4.35 


4.13 


4.33 


4.15 


6 


7 


5.12 


4.77 


5.10 


4.80 


i 5.08 


4.82 


5.06 


4.34 


7 


8 


5. 85 


5.46 


5.83 


5.43 


; 5.80 


5.51 


5.73 


5.53 


8 


9 


6. 58 


6.14 


, 6.56 


6.17 


' 6.53 


6.20 


6.50 


6.22 


9 


10 
11 


7.31 


6. 32 


I 7.23 


6.85 


' 7.25 


6.83 


7.22 i 6 92 


10 


8.04 


7.50 


1 8.01 


7.54 


1 7.93 


7.57 


7.95 


7.61 


11 


12 


8.73 


8.18 


8.74 


8.22 


8.70 


8.26 


8.67 


8.30 


12 


13 


9.51 


8.87 


1 9.47 


8.91 


9.43 


8.95 


9.39 


8.99 


13 


14 


10.24 


9.55 


10.20 


9.59 


; 10.16 


9.64 


10.11 


9.63 


14 


15 


10.97 


10.23 


10.93 


10.23 


■ 10.88 


10.33 


! 10.84 


10.37 


15 


16 


11.70 


10.91 


11.65 


10.96 


11.61 


11.01 


! 11.56 


11.06 


16 


17 


12.43 


11.59 


' 12.33 


11.65 


12.33 


11.70. 


12.28 11.76 


17 


IS 


13.16 


12.23 


; 13.11 


12 . 33 


L3.06 


12.39 


13.00 12.45 


18 


19 


13.90 


12.98 


13.84 


13.02 


; 13.78 


13.03 


13.72 13.14 


19 


20 


14.63 


13.64; 


14.57 


13.70 


i 14.51 


13.77; 


14.45 13.83 


20 

21 


21 


15.36 


14.32^ 


15.30 


14.39 


15.23 


14.46 


15.17 14.52 


22 


16.09 


15.00. 


13.02 


15.07 


15.98 


15.14! 


,15.89 15.21 


22 


23 


16.32 


15.69 


16.75 


15.76 


; 16.63 


15.83 


16.61 15.90 


23 


24 


17.55 


16.37 


17.48 


10.44 


17.41 


16.52 


17.34 16.60 


24 


2o 


IS. 23 


17.05 


13.21 


17.13 


■ 18.13 


17.21 : 


18.06 17.29 


25 


26 


19.02 


17.73 


13.94 


17.81 


'13.86 


17.90, 


18.73 17.93 


26 


27 


19.75 


18.41 • 


19.67 


IS 50 


'19.59 


18.59 


19.50 18.67 


27 


23 


20.43 


18f.l0 


20.39 


19.19 


20.31 


19.27, 


20.23 19.36 


28 


29 


21.21 


19.78, 


21.12 


19.87 


21.04 


19.96; 


20.95 20.05 


29 


30 
31 


21.94 


20.46; 


21.85 


20.56 


21.76 


20.65 ; 


21.67 20.75 


30 
31 


22.67 


21.14, 


22.58 


21.24' 


22.49 


21.34; 


22.39 


21.44 


32 


23.40 


21. 82- 


23.31 


21.93 : 


23.21 


22.03 


23.12 


22.13 


32 


33 


24.13 22.51 ; 


24.04 


22.61 ; 


23.94 


22.72 , 


23.84 


22.82 


33 


34 


24.87 


23.19; 


24.76 


23.30 


24.66 23.40 


24.56 


23.51 


34 


3o 


25.60 


23.87: 


25.49 


23.93 


25.39 24.09 . 


25.23 


24.20 


35 


36 


26.33 


24.55! 


26.22 


24.67: 


26.11 24.73 


26.01 


24.89 


36 


37 


27.06 


25.23 : 


26.95 


25.35] 


26.84 26.47 


26.73 25.59 


37 


38 


27.79 


25.92; 


27.63 


26.04 


27.56 26.16: 


27.45 26.28 


33 


39 


2S.52 


26.60' 


23.41 


26 . 72 i 


23.29 26.85 


23.17 26.97 


39 


40 
41 


29.25, 


27.28; 


29.13 


27.41 ; 


29.01 


27.53 ; 


28.89 
29 . 62 


27.66 


40 
41 


29.99 


27.96, 


29.86 


23.09' 


29.74 


23.22 


23.35 


42 


30.72 28.64: 


.30.59 


23.73 '[ 


30.47 


23.91 ; 


30.34 29.04 


42 


43 


31.45 29.33 


31.32 


29.48 ! 


31.19 


29.60 


31.06 129.74 


43 


44 


32.13 30.01 ' 


32.05 


30.15: 


31.92 


30.29 ; 


31.73 30.43 


44 


45 


32.91 30.69! 


32.73 


30.83 


32.64 


30.98 , 


32.51 131.12 


45 


46 


33.64 31.37; 


33.51 


31.52 


33.37 


31.66 ; 


33.23 31.81 


46 


47 


34.37 32.05' 


34.23 


32.20 


34.09 


32.35 : 


33.95 32.50 


47 


43 35.10 132.74! 


34.96 


32.89 


34.82 


33 . 04 . 


34.67 33.19 


43 


49 


35.84 33.42' 


35.69 


33.57, 


35.54 


33.73 ; 


35.40 33,83 


49 


50 

o 
w 
C 
«i ■ 

5 


36.57 j 34.10 


36.42 


34.26 ; 


30.27 


34.42 ! 


36.12 34.53 


50 

S 
■-> 

■5 


Dep. 


1 
Lat. ' 


Dep. 


Lat 


Dep. 


Lat. I 


Dep. 


Lat. 


47 Deg. 


461 Deg. 

II 


46^ Deg. 


46i Deg. 



TRAVERSE TABLE. 



159 



o 

CD 

51 


43Deg. 


43i Peg. 


43i Deg. 


431 Deg. 


5 

5' 

o 
o 

51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


37.30 


34.78 


37.15 


34.94 


36.99 


35.11 


36.84 


35.27 


52 


38.03 


35.46 


37.88 


35.63 


37.72 


35.79 


37.56 


35.96 


52 


63 


38.76 


36.15 


38.60 


36.31 


38.44 


36.48 


38.29 


36.65 


53 


54 


39.49 


36.83 


39.33 


37.00 


39.17 


37.17 


.39.01 


37.34 


64 


55 


40.22 


37.51 


40.06 


37.69 


39.90 


37.86 


39.73 


38.03 


65 


56 


40.96 


38.19 


40.79 


38.37 


40.62 


38.55 


40.45 


38.72 


66 


57 


41.69 


38.87 


41.52 


39.06 


41.35 


39.24 


41.17 


39.42 


67 


58 


42.42 


39.56 


42.25 


39.74 


42.07 


39.92 


41.90 


40.11 


68 


59 


43.15 


40.24 


42.97 


40.43 


42.80 


40.61 


42.62 


40.80 


69 


60 
61 


43.88 


40.92 


43.70 


41.11 


43.52 


41.30 


43 34 


41.49 


60 
61 


44.61 


41.60 


44.43 


41.80 


44.25 


41.99 


44.06 


42.18 


62 


45.34 


42.28 


45.16 


42.48 


44.97 


42.68 


44.79 


42.87 


62 


63 


46.08 


42.97 


45.89 


43.17 


45.70 


43.37 


45.51 


43.57 


63 


64 


46.81 


43.65 


46.62 


43.85 


46.42 


44.05 


46.23 


44.26 


64 


65 


47.54 


44.33 


47.34 


44.54 


47.15 


44.74 


46.95 


44.95 


65 


C6 


48.27 


45.01 


48.07 


45.22 


4*7.87 


45.43 


47.68 


45.64 


66 


67 


49,00 


45.69 


48.80 


45.91 


48.60 


46.12 


48.40 


46.33 


67 


68 


49.73 


46.38 


49.53 


46.59 


49.33 


46.81 


49.12 


47.02 


6S 


69 


50.46 


47.06 


50.26 


47.28 


50.05 


47.50 


49.84 


47.71 


69 


70 
71 


51.19 


47.74 


.50.99 


47.96 


50.78 


48.18 


50.57 


48.41 


70 

71 


51.93 


48.42 


51.71 


48-65 


51.50 


48.87 


61.29 


49.10 


72 


52.66 


49.10 


52.44 


49.33 


52.23 


49.56 


52.01 


49.79 


72 


73 


53.39 


49.79 


53.17 


50.02 


62.95 


50.25 


.52.73 


50.48 


73 


74 


54.12 


50.47 


53.90 


50.70 


53.68 


50.94 


53.46 


51.17 


74 


75 


54.85 


51.15 


54.63 


51.39 


64.40 


51.63 


54.18 


51.86 


75 


76 


55.58 


51.83 


55.36 


52.07 


55.13 


52.31 


.54.90 


52.55 


76 


77 


56.31 


52.51 


56.08 


52.76 


55.85 


53.00 


55.62 


53.25 


77 


78 


57.05 


53.20 


56.81 


53.44 


56.58 


53.69 


56.34 


53.94 


78 


79 


57.78 


53.88 


57.54 


54.13 


57.30 


54.38 


57.07 


54.63 


79 


80 
81 


58.51 


54.56 


58.27 


54.81 


58.03 


55.07 


57.79 
58.51 


55.32 
56.01 


80 
81 


59.24 


55.24 


59.00 


55.50 


58.76 


65.76 


82 


59.97 


55.92 


59.73 


56.18 


59.48 


56.45 


59.23 


56.70 


82 


83 


60.70 


50.61 


60.45 


56.87 


60.21 


57.13 


.59.96 


57.40 


83 


84 


61.43 


57.29 


61.18 


67.56 


60.93 


57.82 


60.68 


58.09 


84 


85 


62.17 


57.97 


G1.91 


58.24 


61.60 


58.51 


61.40 


58.78 


85 


88 


62.90 


58.65 


62.64 


58.93 


62.38 


59.20 


62.12 


59.47 


86 


87 


63.63 


59.33 


63.37 


59.61 


63.11 


59.89 


62.85 


60.10 


87 


88 


64.36 


60.02 


64.10 


60.30 


63.83 


60.58 


63.57 


60.85 


88 


89 


65.09 


60.70 


64.82 


60.98 


64.56 


61.26 


64.29 


61.54 


89 


90 
91 


65.82 


01.38 


65.55 
'66'. 28 


61.67 


65.28 


61.95 


65.01 


62.24 


90 
91 


66.55 


62.06 


62.35 


66.01 


62.64 


65.74 


62.93 


92 


67.28 


02.74 


67.01 


03.04 


66.73 


63.33 


66.46 


63.62 


92 


93 


68.02 


63.43 


67.74 


63.72 


07.46 


64.02 


67.18 


64.31 


93 


94 


68.75 


64.11 


68.47 


64.41 


68.19 164.71 


67.90 


65.00 


94 


95 


09.48 


64.79 


09.20 


65.09 


68.91 


65.39 


68.62 


65.69 


95 


96 


70.21 


65.47 


69.92 


65.78 


69 . 64 


66.08 


69.35 


66.39 


96 


97 


70.94 


66.15 


70.65 


66.46 


70.30 


66.77 


70.07 


67.08 


97 


98 


71.67 


66.84 


71.37 


67.15 


71.09 


67.46 


70.79 


67.77 


98 


99 


72.40 


67.52 


72.11 


67.83 


71.81 


68.15 


71.51 


68.46 


99 


100 

o 

a 
a 
«-> 
to 


73.14 


88.20 


72.84 


68 . 52 


72.54 


68.84 


72 . 24 


69.15 


100 

o 
o 

*-> 

03 

Q 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


47 Deg. 


.461 Deg. 

1 


46-^- Deg. 


464 Deg. 



30 



160 



TRAVEKSE TABLE. 



5 

so 

p 

3 
O 
p 


44 Deg. 


44i Deg. 


44i Deg. 


44| Deg. 


45 Deg. 


O 

55' 

<-► 
p 

3 
o 
9 


Lai. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


0.72 


0.69 


0.72! 0.70 


0.71 


0.70 


0.71 


0.71 


0.71 


0.71 


1 


2 


1.44 


1.39 


1.43 


1.40 


1.43 


1.40 


1.42 


1.41 


1.41 


1.41 


2 


3 


2.16 


2.08 


2.15 


2.09 


2.14 


2.10- 


2.13 


2.11 


2.12 


2.12 


3 


4 


2.88 


2.78 


2.87 


2.79 


2.85 


2.80 


2.84 


2.82 


2.83 


2.83 


4 


5 


3.60 


3.47 


3.58 


3.49 


3.57 


3.50 


3.55 


3.52 


3.54 


3.54 


5 


6 


4.32 


4.17 


4.30 


4.19 


4.28 


4.21 


4.26 


4.22 


4.24 


4.24 


6 


7 


5.04 


4.86 


5.01 


4.88 


4.99 


4.91 


4.97 


4.93 


4.95 


4.95 


7 


8 


5.75 


5.56 


5.73 


5.58 


5.71 


5.61 


5.68 


5.63 


5.66 


5.66 


8 


9 


6.47 


6.25 


6.45 


6. 28 


G.42 


6.31 


6.39 


6.34 


6.36 


6.36 


9 


lU 
11 


7.19 
7.91 


6.95 


7.16 


6.98 


7.13 


7.01 
7.71 


7.10 


7.04 


7.07 


7.07 


10 
11 


7.64 


7.88 


7,68 


7.85 


7.81 


7.74 


7.78 


7.78 


12 


8.63 


8.04 


8.60 


8.37 


8.56 


8.41 


8.52 


8.45 


8.49 


8.49 


12 


13 


9.35 


9.03 


9.31 


9.07 


9.27 


9.11 


9.23 


9.15 


9.19 


9.19 13 


14 


10.07 


9.73 


10,03 


9.77 


9.99 


9.81 


9.94 


9.86 


9.90 


9.90 14 


15 


10.79 


10.42 


10.74 10.47 


10.70 


10.51 


10.65 


10.56 


10.61 


10.61 


15 


16 


11.51 


11.11 


11.46 11.16 


11.41 


11.21 


11.36 


11.26 


11.31 


11.31 


16 


17 


12.23 


11.81 


12.18 11.86 


12.13 


11.92 


12.07 


11.97 


12.02 


12.02 


17 


18 


12.95 


12.50 


12.89 12.56 


12.84 


12.62 


12.78 


12.67 


12.73 


12.73 


18 


19 


13.67 


13.20 


13.61 


13.26 


13.. 55 


13.32 


13.49 


13.38 


13.43 


13.43 


19 


20 
21 


14.39 


13.89 


14.33 


13.96 


14.26 


14.02 


14.20 


14.08 


14.14 


14.14 


20 
21 


15.11 


14.59 


15.04 


14.65 


14.98 


14.72 


14.91 


14.78 


14.85 


14.85 


22 


15.83 


15.28 


15.76 


15.35 


15.69 


15.42 


15.62 


15.49 


15.56 


15.56 


22 


23 


16.54 


15.98 


16.47 


16.05 


16.40 


16.12 


16.33 


16.19 


16.26 


16.26 


23 


24 


17.26 


16.67 


17.19 


16.75 


17.12 


16.82 


17.04 


16.90 


16.97 


16.97 


24 


25 


17.98 


17.37 


17.91 


17.44 


17.83 


17.52 


17.75 


17.60 


17.68 


17.68 


25 


26 


18.70 


18.06 


18.62 


18.14 


18.54 


18.22 


18.46 


18.30 


18.38 


18.38 26 


27 


19.42 


18.76 


19.34 


18.84 


19.26 


18.92 


119.17 


19.01 


19.09 


19.09 27 


28 


20.14 


19.45 


20.06 19.541 


19.97 


19.63,19.89 


19.71 


19.80 


19.80 28 


29 


20.86 


20.15 


20.77 


20.24 


20.68 


20.33 


20.60 


20.42 


20.51 


20.51 29 


30 


21.58 


20.84 


21.49 


20.93 


21.40 
22.11 


21.03 
21.73 


21.31 
,22.02 


21.12 


21.21 


21.21 30 
21.92 31 


31 


22.30 


21.53 


22.21 


21.63 


21.82 


21.92 


32 


23.02 


22.23 


22.92 


22.33 


22.82 


22.43 


22.73 


22.53 


22.63 


22.63 


32 


33 


23.74 


22.92 


23.64 


23.03 


23.54 


23.13 


23.44 


23.23 


23.33 


23.33 


33 


34 


24.46 


23.62 


24.35 


23.72 


24.25 


23.83 


'24.15 


23.94 


24.04 


24.04 


34 


35 


25.18 


24.31 


25.07 


24.42 


24.96 


24.53 


24.86 


24.64 


24.75 


24.75 


35 


36 


25.90 


25.01 


25.79 


25.12 


25.68 


26.23 


125.. 57 


25.34 


25.46 


25.46!36l 


37 


26.62 


25.70 


26.50 


25.82 


26.39 


25.93 


26.28 


26.05 


26.16 


26.16 


37 


38 


27.33 


26.40 


27.22 26.52 


27.10 


26.63 


[26.99 


26.75 


26.87 


26.87 


38 


39 


28.05 


27.09 


27.94 27.21 


27.82 


27.34 


27.70 


27.46 


27.58 


27.58 


39 


40 
41 


28.77 
29.49 


27.79 


28.65 27.91 


28 . 53 


28.04 


28.41 
(29.12 


28.16 


28.28 


28.28 


40 
41 


28.48 


29.37 


28.61 


29.24 


28.74 


28.86 


28.99 


28.99 


42 


30.21 


29.18 


30.08 


29.31 


29.96 


29.44 


,29.83 


29.57 


29.70 


29.70 


42 


43 


30.93 


29.87 


30.80 


30.00 


30.67 


30.14 


30.54 


30.27 


30.41 


30.41 


43 


44 


31.65 


30.56 


31.52 


30.70 


31.38 


30.84 


31.25 


30.98 


31.11 


31.11 


44 


45 


32.37 


31.26 


32.23 


31.40 


.32.10 


31.54 


31.96 


31. 6S 


31.82 


31.82 


45 


46 


33.09 


31.95 


32.95 


32.10 


32.81 


32.24 


32.67 


32.38 


32.53J32.53 


46 


47 


33.81 


32.05 


33.67 


32.80 


33.. 52 


32.94 


33.38 


33.09 


33.23 


33.23 


47 


48 


34.53 


33.34 


34.38 


33.49 


34.24 


33.64 


34.09 


33.79 


.33.94 


33.94 


48 


49 


35.25 


34.04 


3.5.10 


34.19 


34.95 


34.34 


34.80 


34.50 


34.65 


34.65 


49 


50 

§ 

5 


35.97 
Dep. 


34.73 


35.82 


34.89 


35.66 


35.05 


35.51 


35.20 


36.36 


35.36 


50 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


6 

c 

•a 

o 


46 Deg. 


45| Deg. 


45^ Deg. 


45k Deg. 


45 De^., 



TSAVBBSE TABLE 



161 



Ul' 

p> 

r> 

9 

51 
52 
53 
54 
55 
66 
57 
58 
69 
60 

61 
62 
63 
64 
65 
66 
67 
68 
69 
70 

71 
72 
73 

74 
.75 
76 
77 
78 
79 
80 

81 
82 
83 
84 
85 
86 
87 
88 
89 
90 

91 
92 
93 
94 
95 
96 
97 
98 
99 
100 

8 

B 

2 
Q 


44Deg. 


44i Deg. 


44i Deg, 


44| Deg. 


45 Deg. 




CO 

r* 
P» 

3 


9 

61 

62 

63 

54 

65 

56 

57 

58 

59 

60 

61 
62 
63 
04 
65 
66 
67 
68 
69 
70 

71 
72 
73 
74 
75 
76 
77 
78 
79 
80 

81 
82 
83 
84 
85 
86 
87 
88 
89 
90 

91 
92 
93 
94 
95 
96 
97 
98 
99 
100 

S 

d 

5 


Lat. 


Dep. 


Lat. 


Dep. 


Lai. 


Dep. 


Lat. 


Dep. 

35.90 
36.61 
37.31 

38.02 
38,721 
39,42; 
40.13 
40.83 
41.54 
42.24 


Lat. 


Dep. 


36.69 
37.41 
38.12 
38.84 
39.56 
40.28 
41.00 
41.72 
42.44 
43.16 


35.43 
36.12 
36.82 
37.51 
38.21 
38.90 
39.60 
40.29 
40.98 
41.68 


36.53 
37.25 
37.96 
38.68 
39.40 
40.11 
40.83 
41.55 
42.26 
42.98 


35,59 
36.29 
36.98 
37.68 
38.38 
39.08 
39.77 
40.47 
41.17 
41.87 


36.38 
37.09 
37.80 
38.52 
39.23 
39.94 
40.66 
41.37 
42.08 
42.79 
43.51 
44.22 
44.93 
45.65 
46.36 
47.07 
47.79 
48.60 
49.21 
49.93 

60.64 
51.35 
52.07 
52.78 
53.49 
54.21 
54.92 
55.63 
56.35 
57.06 


35.75 
36.45 
37.15 
37.85 
38.55 
39,25 
39.95 
40.65 
41.35 
42.05 


36.22 
36.93 
37.64 
38.35 
39.06 
39.77 
40.48 
41.19 
41.90 
42.61 


36.06 
36,77 
37,48 
38,18 
38,89 
39.60 
40.31 
41,01 
41.72 
42,43 


36.06 
36.77 
37,48 
38.18 
38.89 
39.60 
40,31 
41,01 
41.72 
42.43 


43. 8S 
44.60 
45.32 
46.04 
46.76 
47.48 
48,20 
48.92 
49.63 
50.35 


42.37 
43.07 
43.76 
44.46 
45.15 
45.85 
46.54 
47.24 
47.93 
48.63 


43.69 
44.41 
45.13 
45.84 
46.56 
47.28 
47.99 
48.71 
49.42 
50.14 


42.57 
43.26 
43.96 
44.66 
45,36 
46.05 
46.75 
47.45 
48.15 
48.85 


42.76 
43.46 
44.16 
44.86 
45.56 
46.26 
46.96 
47.66 
48.36 
49.06 


43.32 
44.03 
44,74 
45.45 
46.16 
46.87 
47.58 
48,29 
49.00 
49.71 


42.94 
43.65 
44.35 
45.06 
45.76 
46.46 
47.17 
47.87 
48.58 
49.28 


43,13 
43.84 
44.55 
45.25 
45.96 
46.67 
47.38 
48.08 
48.79 
49.50 


43.13 

43.84 
44.55 
45.25 
45.96 
46.67 
47.38 
48.08 
48.79 
49.50 

50.20 
50.91 
51.62 
52.33 
53.03 
53.74 
54.45 
55.15 
55,86 
56.57 


51.07 
51.79 
52.51 
53.23 
53.95 
54.67 
55.39 
56.11 
56.83 
57.55 


49.32 
50.02 
50.71 
51.40 
52.10 
,52.79 
53.49 
54. 18 
54.88 
55.57 
56.27 
56.96 
57.66 
58.35 
59.05 
59.74 
60.44 
61.13 
61.82 
62.52 


50.86 
51.67 
52.29 
53.01 
53.72 
54.44 
55.16 
55.87 
56.59 
.57.30 


49.54 
50.24 
50.94 
51.64 
52.33 
53.03 
.53.73 
54.43 
55.13 
55.82 


49.76 
50.47 
51.17 
51.87 
,52.57 
53.27 
53.97 
54-67 
55.37 
56.07 


50.42 
51.13 
51-84 
52.55 
53.26 
53,97 
54.68 
;55.39 
156.10 
56.81 


49.98 
50.69 
51.39 
52.10 
52.80 
53.51 
54.21 
54.91 
55.62 
56.32 


50.20 
50.91 
51.62 
52.33 
53.03 
53.74 
54.45 
55.15 
55.86 
56.67 

57.28 
57.98 
58.69 
59.40 
60.10 
60.81 
61.52 
62.23 
62.93 
63.64 


58.27 
58.99 
59.71 
60.42 
61.14 
61.86 
62.58 
63.30 
64.02 
64.74 


58.02 
58.74 
59.45 
60.17 
60.89 
61.60 
62.32 
63.03 
63.75 
64.47 


56.52 
57.22 
57.92 
58.61 
59.31 
60.01 
60.71 
61.41 
62,10 
62.80 


57.77 
58.49 
59.20 
59.91 
60.63 
61.34 
62.05 
62,77 
63.48 
64.19 

64.91 
65.62 
66.33 
67.05 
67.76 
68.47 
69.19 
69.90 
70.61 
71.33 


56.77 
57.47 
58.18 
58.88 
59.58 
60.28 
60.98 
61.68 
62.. 38 
63.08 


57.52 
58.24 
•68.95 
159.66 
160.37 
i6l.08 
161.79 
162.50 
163.21 
63.92 
64.63 
65.34 
66.05 
66.76 
67.47 
68.18 
68.89 
69.60 
70.31 
71.02 


57.03 
57.73 
58.43 
59.14 
59.84 
60.65 
61.25 
61.95 
62,66 
63.36 


57.28 
57.98 
58.69 
59:40 
60.10 
60.81 
61.52 
62.23 
62.93 
63,64 

64.35 
65.05 
65.76 
66.47 
67.18 
67.88 
68.59 
69.30 
70.00 
70.71 


65.46 
66.18 
66.90 
67.62 
68.34 
69.06 
69.78 
70.50 
71.21 
71.93 


63.21 
63.91 
64.60 
65.30 
65.99 
66.69 
67.38 
68.08 
68.77 
69.47 


65.18 
65.90 
66.62 
67.33 
68.05 
68.76 
69.48 
70.20 
70.91 
71.63 


63.50 
64.20 
64.89 
65.59 
66.29 
66.99 
67.69 
68.38 
69.08 
69.78 


63.78 
64.48 
65.18 
65.89 
66.59 
67.29 
67.99 
68.69 
69.39 
70.09 


64.07 
64.77 
65-47 
66.18 
66.88 
67,59 
68,29 
68.99 
69.70 
70.40 


64.35 
66,05 
65,76 
66,47 
67.18 
67.88 
68.59 
69.30 
70.00 
70.71 

Dep. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Lat. 


46 Deg. 


45| Deg. 


45 i Deg. 


45i Deg. 


45 Deg. 



Phy 



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Geology 



Dana's Geological Story Briefly Told 

By James D. Dana. Cloth, i2mo, 302 pages . . . $1.15 

A new edition of this popular work for beginners in the study and for the general 
reader. The book has been entirely rewritten, and improved by the addition of many 
new illustrations and interesting descriptions of the latest phases and discoveries of 
the science. In contents and dress it is an attractive volume either for the reader or 
student. 

Dana's New Text-Book of Geology 

By James D. Dana. Cloth, i2mo, 422 pages . . . $2.00 

A text-book for classes in secondary schools and colleges. This standard work 
has been thoroughly revised and considerably enlarged and freshly illustrated to 
represent the latest demands of the science. 

Dana's Manual of Geology 
By James D. Dana. 
Cloth, 8vo, 1087 pages. 1575 Illustrations .... $5.00 

Fourth revised edition. This great work was thoroughly revised and entirely 
rewritten under the direct supervision of its author, just before his death. It is recog- 
nized ^s a standard authority in the science both in Europe and America, and is used 
as a manual of instruction in all the higher institutions of learning. 

Le Conte's Compend of Geology 

By Joseph Le Conte, LL.D. Cloth, i2mo, 399 pages . $1.20 

Designed for high schools, academies and all secondary schools. 

Steele's Fourteen Weeks in Geology 

By J. DoRMAN Steele, Ph.D. Cloth, i2mo, 280 pages . $1.00 
A popular book for elementary classes and the general reader. 

Andrews's Elennentary Geology 

By E. B. Andrews, LL.D. Cloth, i2mo, 283 pages . $1.00 

Adapted for elementary classes. Contains a special treatment of the geology of 
the Mississippi Valley. 

Nicholson's Text-Book of Geology 

By H. A. Nicholson, M.D. Cloth, i2mo, 520 pages . $1.05 

A brief course for higher classes and adapted for general reading, 

Willianas's Applied Geology 

By S. G. Williams, Ph.D. Cloth, i2mo, 386 pages . . $1.20 

A treatise on the industrial relations of geological structure; and on the nature, 
occurrence, and uses of substances derived from geological sources. 



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Text-Books in Astronomy 

Bowen's Astronomy by Observation 
By Eliza A. Bowen. 

Boards, quarto, 94 pages. Colored Maps and Illustrations $1 .00 

An elementary text-book for schools, and especially adapted for use 

as an atlas to accompany any other text-book in astronomy. Careful 

directions are given when, how and where to find the heavenly bodies, 

and the quarto pages admit star maps and views on a large scale. 

Gillet and Rolfe's Astronomies 
By T. A. Gillet and W. J. Rolfe. 

First Book in Astronomy. Short Course. 220 pages . . $1.00 

Astronomy. 415 pages 1.40 

These books have been prepared by practical teachers and contain 

nothing beyond the comprehension of pupils in secondary schools, 

Lockyer's Astronomies 

By J. N. LocKYER, F.R.S. 

Astronomy. (Science Primer Series.) 136 pages . 35 cents 

Elementary Lessons in Astronomy, 312 pages . . . $1.22 

The aim throughout these books is to give a connected view of the 

whole subject rather than to discuss any particular parts of it, and to supply 

facts and ideas founded thereon, to serve as a basis for subsequent study. 

Ray's New Elements of Astronomy 
By Selim H. Pe-\body, Ph.D., LL.D. 

Cloth, i2mo, 352 pages $1 .20 

The elements of astronomy, with numerous engravings and star maps. 
In the revised edition, the scope and method of the original is retained, 
with the addition of all the results of established discovery. The book 
treats of the facts, principles, and processes of the science, presuming 
only that the pupil is acquainted with the simplest principles of mechanics 
and physics. 

Steele's New Descriptive Astronomy 

By J. D0R3IAN Steele, Ph.D. Cloth, i2mo, 338 pages , $1.00 
This book is written in the same interesting and popular manner as 
other books of the Steele Series, and is intended for the inspiration of 
youth rather than for the information of scientific scholars. The book 
conforms to the latest discoveries and approved theories of the science. 
It supplies an adequate course in astronomy for all secondary schools 
and college preparatory classes. 



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Physical Geography 



Appletons' Physical Geography 

By John D. Quackenbos, John S. Newberry, Charles H. 
Hitchcock, W. Le Conte Stevens, Wm. H. Dall, Henry 
Gannett, C. Hart Merriam, Nathaniel L. Britton, 
George F. Kunz and Lieut, Geo. M. Stoney. 

Cloth, quarto, 140 pages ....... $1 .60 

Prepared on a new and original plan. Richly illustrated with engrav- 
ings, diagrams and maps in color, and including a separate chapter on 
the geological history and the physical features of the United States. 
The aim has been to popularize the study of Physical Geography by 
furnishing a complete, attractive, carefully condensed text-book. 

Cornell's Physical Geography 

Boards, quarto, 104 pages . . . , . . $1,12 

Revised edition, with such alterations and additions as were found 
necessary to bring the work in all respects up to date. 

Hinman's Eclectic Physical Geography 

Cloth, i2mo, 382 pages $1.00 

By Russell Hinman. A model text-book of the subject in a new 
and convenient form. It embodies a strictly scientific and accurate 
treatment of Physiography and other branches of Physical Geography. 
Adapted for classes in high schools, academies and colleges, and for 
private students. The text is fully illustrated by numerous maps, 
charts, cuts and diagrams. 

Guyot's Physical Geography 

Cloth, quarto, 124 pages $1.60 

By Arnold Guyot. Thoroughly revised and supplied with newly 
engraved maps, illustrations, etc. A standard work by one of the ablest 
of modern geographers. All parts of the subject are presented in their 
true relations and in their proper subordination. 

Monteith's New Physical Geography 

Cloth, quarto, 144 pages ....... $1.00 

An elementary work adapted for use in common and grammar schools, 
as well as in high schools. 



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STORER AND LINDSAY'S 

Elementary Manual of Chemistry 

By F. H. STORER, S.B., A.M., and W. B. LINDSAY, A.B., B.S. 
Cloth, 12mo, 453 pages. Illustrated. Price, $1.20 

This work is the lineal descendant of the " Manual of 
Inorganic Chemistry" of Eliot and Storer, and the ''Ele- 
mentary Manual of Chemistry " of Eliot, Storer and Nichols. 
It is in fact the last named book thoroughly revised, 
rewritten and enlarged to represent the present condition 
of chemical knowledge and to meet the demands of American 
teachers for a class book on Chemistry, at once scientific 
in statement and clear in method. 

The purpose of the book is to facilitate the study and 
teaching of Chemistry by the experimental and inductive 
method. It presents the leading facts and theories of the 
science in such simple and concise manner that they can 
be readily understood and applied by the student. The 
book is equally valuable in the classroom and the laboratory. 
The instructor will find in it the essentials of chemical 
science developed in easy and appropriate sequence, its 
facts and generalizations expressed accurately and scientifi- 
cally as well as clearly, forcibly and elegantly. 



" It is safe to say that no text-book 

has exerted so wide an influence 
on the study of chemistn' in this 
countn^ as this work, originally 
written by Eliot and Storer. Its 
distinguished authors were leaders 
in teaching Chemistr}' as a means 
of mental training in general edu- 
cation, and in organizing and per- 
fecting a system of instructing 
students in large classes by the 
experimental method. As revised 
and improved by Professor Nichols, 
it continued to give the highest 
satisfaction in our best schools and 
colleges. After the death of Pro- 
fessor Nichols, when it became 



necessary to revise the work again, 
Professor Lindsay, of Dickinson 
College, was selected to assist Dr. 
Storer in the work. The present 
edition has been entirely rewritten 
by them, following throughout the 
same plan and arrangement of the 
previous editions, which have been 
so highly approved by a generation 
of scholars and teachers. 

" If a book, like an individual, 
has a history, certainly the record 
of this one, covering a period of 
nearly thirty years, is of the highest 
and most honorable character," 
— From The American Jotirnal of 
Science, 



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Chicago 



Burnet's Zoology 

FOR 
HIGH SCHOOLS AND ACADEMIES 

BY 

MARGARETTA BURNET 

Teacher of Zoology, Woodward High School, Cincinnati, O. 

Cloth, 12mo, 216 pages. Illustrated. Price, 75 cents 



This new text-book on Zoology is intended for classes 
in High Schools, Academies, and other Secondary Schools. 
While sufficiently elementary for beginners in the study it is 
full and comprehensive enough for students pursuing a 
regular course in the Natural Sciences. It has been prepared 
by a practical teacher, and is the direct result of school-room 
experience, field observation and laboratory practice. 

The design of the book is to give a good general knowl- 
edge of the subject of Zoology, to cultivate an interest in 
nature study, and to encourage the pupil to observe and to 
compare for himself and then to arrange and classify his 
knowledge. Only typical or principal forms are described, 
and in their description only such technical terms are used 
as are necessary, and these are carefully defined. 

Each subject is fully illustrated, the illustrations being 
selected and arranged to aid the pupil in understanding the 
structure of each form. 



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Laboratory Physics 



Hammers Observation Blanks In Physics 

By William C. A. Hammel, Professor of Physics in 
Maryland State School. Boards, Quarto, 42 pages. 
Illustrated. 30 cents 

These Observation Blanks are designed for use as a 
Pupil's Laboratory Manual and Note Book for the first 
term's work in the study of Physics. They combine in 
convenient form descriptions and illustrations of the appa- 
ratus required for making experiments in Physics, with 
special reference to the elements of Air, Liquids, and Heat; 
directions for making the required apparatus from simple 
inexpensive materials, and for performing the experiments, 
etc. The book is supplied with blanks for making drawings 
of the apparatus and for the pupil to record what he has 
observed and inferred concerning the experiment and the 
principle illustrated. 

The experiments are carefully selected in the light of 
experience and arranged in logical order. The treatment 
throughout is in accordance with the best laboratory practice 
of the day. 

Hon. W. T. Harris, U. S. Commissioner of Education, 
says of these Blanks: 

" I have seen several attempts to assist the work of 
pupils engaged in the study of Physics, but I have never 
seen anything which promises to be of such practical assist- 
ance as Hammel's Observation Blanks." 



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Standard Text-Books in Botany 



Gray's How Plants Grow. (Introductory Book) . 

Gray's How Plants Behave 

For Beginners in Primary and Intermediate Schools. 
Gray's Lessons in Botany. (Revised) .... 
Gray's Field, Forest and Garden Botany. (Flora) . 
Gray's School and Field Botany. (The Standard Text-Book) 

For Students in High Schools, Academies and Seminaries. 

Gray's Manual of Botany. (Flora) 

Gray's Lessons and Manual. (In one volume) . 

For Advanced Students, Teachers, and Practical Botanists. 
Coulter's Botany of the Rocky Mountains 

A flora adapted to the mountain section of the United States. 

Gray and Coulter's Text-Book of Western Botany . 

Being Gray's Lessons and Coulter's Manual bound in one volume. 

Gray's Structural Botany 

Goodale's Physiological Botany 

Dana's Plants and their Children 

Herrick's Chapters on Plant Life 

Hooker's Botany. (Science Primer Series) .... 
Hooker's Child's Book of Nature. Part I. Plants 
Steele's Fourteen Weeks in Botany .... 
Wood's How to Study Plants 

Same as above work, with added chapters on Physiological and Sys- 
tematic Botany. 

Wood's Lessons in Botany. (Revised) .... 

Wood's New Annerican Botanist and Florist. (Revised) . 

Wood's Descriptive Botany ...... 

Beinp- the flora of the American Botanist and Florist. 

Wood's Class Book of Botany 

A standard work for Advanced Classes and for the Student's Library. 

Younnans's First Book in Botany 

Younaans's Descriptive Botany . . . 
Bentley's Physiological Botany 

A sequel to Youmans's Descriptive Botany. 

Willis's Practical Flora 

A valuable supplementary aid to any text-book in the study of Botany. 



80 cents 
54 cents 

94 cents 
$1.44 
$1.80 

$1.62 
$2.16 

$1.62 

$2.16 

$2.00 

$2.00 

65 cents 

60 cents 

35 cents 

44 cents 

$1.00 

$1.00 

90 cents 
$1.75 
$1.25 

$2.50 

64 cents 
$1.20 
$1.20 

$1.50 



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Physiology and Hygiene 



Kellogg's First Book in Physiology and Hygiene 

Cloth, i2mo, 174 pages 40 cents 

Kellogg's Second Book in Physiology and Hygiene 

Cloth, i2mo, 2gi pages 80 cents 

These two books constitute an entirely new and well graded series 
for the study of Physiolog}' and Hygiene in schools. The subjects are 
treated in a natural and logical order and arranged in a form suitable for 
class instruction. The important subjects of sanitation and temperance 
are thoroughly treated from a scientific and physiological standpoint. 

Snnith's Prinner of Physiology and Hygiene 

Cloth, i2mo, 174 pages 30 cents 

Snnith's Elementary Physiology and Hygiene 

Cloth, i2mo, 225 pages ....... 50 cents 

A complete and symmetrical series in which the important facts of 

Physiology and Hygiene are presented in an interesting manner. The 

Primer is designed for beginners in the study and the second book for 

classes in the intermediate grades. 

Steele's Hygienic Physiology 

Cloth, i2mo, 400 pages $1 .00 

This standard text-book has been thoroughly revised and consider- 
ably enlarged. It contains all the excellent and popular features that 
have given Dr. Steele's Science Series such wide use in schools throughout 
the country. 

The Same, abridged. Cloth, i2mo, 192 pages . . 50 cents 

Tracy's Essentials of Anatomy, Physiology and Hygiene 

Cloth, i2mo, 345 pages $1.00 

A practical, thorough and scientific text-book of an advanced grade 

for the use of classes in High Schools, Academies, Normal Schools, and 

for private students. 

Johonnot and Bouton's How We Live 

Cloth, i2mo, 178 pages 40 cents 

An elementary text-book for beginners in which special attention is 
given to the laws of Hygiene. 

Walker's Health Lessons 

Cloth, i2mo, 194 pages 48 cents 

A book for beginners, presenting the subjects in an interesting and 

readable form suitable for supplementary readings. 



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lb D 78 



